HBEC2503 Mathematics in Early Childhood Edu

HBEC2503 MATHEMATICS IN EARLY CHILDHOOD EDUCATION Nik Noraini Nik Abu Bakar Prof Dr Lim Tick Meng

Project Directors:

Prof Dr Mansor Fadzil Prof Dr Widad Othman Open University Malaysia

Module Writers:

Nik Noraini Nik Abu Bakar Prof Dr Lim Tick Meng Open University Malaysia

Moderator:

Dr Azhar Md Adnan Open University Malaysia

Developed by:

Centre for Instructional Design and Technology Open University Malaysia

Printed by:

Meteor Doc. Sdn. Bhd. Lot 47-48, Jalan SR 1/9, Seksyen 9, Jalan Serdang Raya, Taman Serdang Raya, 43300 Seri Kembangan, Selangor Darul Ehsan

First Edition, December 2014 Copyright © Open University Malaysia (OUM), December 2014, HBEC2503 All rights reserved. No part of this work may be reproduced in any form or by any means without the written permission of the President, Open University Malaysia (OUM).

Table of Contents Course Guide

ix-xiv

1.1 1.2

Misconceptions of Early Childhood Mathematics 2 Theories in Early Childhood Learning of Mathematics 4 1.2.1 PiagetÊs Theory of Child Development 4 1.2.2 Lev Vygotsky and the Social Development Theory 8 1.2.3 Six Stage Theory of Zoltan P. Dienes 9 1.2.4 Jerome BrunerÊs Three Modes of Representation 10 Good Practices in Early Childhood Mathematics 12 1.3.1 NCTM Recommendations for Childhood Mathematics 13 1.3.2 Recommendations from the Australian Association of 13 Mathematics Teachers Content Standards in Early Childhood Mathematics 15 1.4.1 Kurikulum Standard Prasekolah Kebangsaan (KSPK) 15 1.4.2 Hawaii Preschool Mathematics Content Standard 19 1.4.3 NCTMÊs and NAEYCÊs Proposed Preschool 20 Mathematics Content Standard

1.3

1.4

Summary Key Terms References

22 22 22

2.1 2.2

25 26

The Role of Language and Language of Mathematics Developing Mathematical Language Using Songs and Nursery Rhymes 2.3 Useful Words to Guide Learning of Early Mathematical Concepts 2.4 Mathematical Talk and Early Mathematical Concepts 2.4.1 Examples of Mathematical Talk 2.5 Integrating Mathematics into ChildrenÊs Play Summary Key Terms

3.1 3.2

Classification 3.1.1 The Understanding of Sets Matching

31 32 36 37 38 39

41 43 44

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TABLE OF CONTENTS

3.2.1 Challenging Matching Task 46 3.3 Sorting, Categorising and Grouping 3.4 Ordering or Seriation 3.5 Informal Learning at Home or at School 3.6 Assessment on Classifying, Matching, Sorting and Ordering Summary Key Terms References

4.1 4.2

48 49 51 52 55 55 55

The Importance of Patterns Types of Patterns 4.2.1 Repeating Patterns 4.2.2 Growing Patterns 4.2.3 Shrinking Patterns 4.3 Creating Pattern Activities 4.3.1 People Patterns 4.3.2 Patterns with Concrete Objects and Pattern Cards 4.3.3 Creating Patterns Using Music 4.3.4 Creating Patterns in Art 4.4 Number Patterns 4.5 Children as Creator of Patterns 4.6 Assessment on learning of Patterns 4.6.1 Formative Assessment 4.6.2 Summative Assessment Summary Key Terms References

57 58 58 60 60 62 62 63 64 65 67 68 69 70 70 71 71 71

5.1

73 74 76 77 77 78 78

5.2

Developing Number Sense 5.1.1 Understanding Number Sense 5.1.2 Number Sense Ability in Children Multiple Conception of Place Value 5.2.1 Unique Features of Base Ten System 5.2.2 Place Value 5.2.3 Base 10 Block Concepts

5.2.4 2-Digit Number Concepts Summary Key Terms References

81 85 85 86

TABLE OF CONTENTS

6.1

Classroom Experience with Problem Solving 6.1.1 Everyday Opportunities to Experience Problem Solving 6.1.2 Guidance and Strategies to Foster Problem Solving Skills in Young Children 6.2 Addition and Subtraction 6.3 Multiplication and Division Summary Key Terms References

7.1

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v

88 89 89 90 93 97 98 98

Usefulness of Charts 7.1.1 Varieties of Charts and Graphs 7.2 Early Experience with Graphing 7.2.1 Mapping Techniques, from Pre-Graphic to Early Graphic 7.3 Probability Summary Key Terms References

109 112 112 112

8.1 Measurement 8.2 Principles of Measurement 8.3 Formal Approaches and Techniques of Measurement 8.4 Volume, Weight, Length and Temperature 8.5 Informal Measurement: The Concept of Time Summary Key Terms

115 117 120 123 129 131 131

9.1 9.2 9.3

133 136 138

Managing the Learning Environment Social And Emotional Structures of Learning Instructional Strategies for Enhancing Mathematics Learning Environment

Summary Key Terms References

100 101 105 108

150 150 150

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TABLE OF CONTENTS

10.1 Assessment of Mathematics in Early Childhoohd 10.2 Role of Assessment 10.3 Assessment Strategy 10.4 Assessment Results Analysis Summary Key Terms References

152 158 164 167 168 168 168

COURSE GUIDE

COURSE GUIDE

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ix

COURSE GUIDE DESCRIPTION You must read this Course Guide carefully from the beginning to the end. It tells you briefly what the course is about and how you can work your way through the course material. It also suggests the amount of time you are likely to spend in order to complete the course successfully. Please keep on referring to the Course Guide as you go through the course material as it will help you to clarify important study components or points that you might miss or overlook.

INTRODUCTION is one of the courses offered by Faculty of Education at Open University Malaysia (OUM). This course is worth 3 credit hours and should be covered over eight to 15 weeks.

COURSE AUDIENCE This course is offered to all learners undertaking the Bachelor in Childhood Education with Honours. As an open and distance learner, you should be acquainted with learning independently and being able to optimise the learning modes and environment available to you. Before you begin this course, please ensure that you have the right course material, and understand the course requirements as well as how the course is conducted. As an open and distance learner, you should be acquainted with learning independently and being able to optimise the learning modes and environment available to you. Before you begin this course, please ensure that you have the right course material, and understand the course requirements as well as how the course is conducted.

STUDY SCHEDULE It is a standard OUM practice that learners accumulate 40 study hours for every credit hour. As such, for a three-credit hour course, you are expected to spend 120 study hours. Table 1 gives an estimation of how the 120 study hours could be accumulated.

x

 COURSE GUIDE

Estimation of Time Accumulation of Study Hours

Briefly go through the course content and participate in initial discussion

3

Study the module

60

Attend 3 to 5 tutorial sessions

10

Online participation

12

Revision

15

Assignment(s), Test(s) and Examination(s)

20

COURSE OUTCOMES By the end of this course, you should be able to: 1. Summarise the sequential development of mathematical concepts; 2. Describe how the development of mathematical concepts promotes childrenÊs thinking skills; 3. Develop strategies which promote thinking and problem-solving skills in children; 4. Utilise observation and assessment as a basis for planning learning activities; and 5.

Create, evaluate and select developmentally appropriate materials, equipment and environment to support the attainment of mathematical concepts.

COURSE SYNOPSIS This course is divided into 10 topics. The synopsis for each topic can be listed as follows: gives an overview of the various misconceptions about early childhood mathematics education by explaining the theories of learning of mathematics in relation to the learning of early childhood. This topic also suggests, good practices that help to improve the quality of early mathematics education and outlines the standard contents for early childhood mathematics.

COURSE GUIDE

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xi

defines the role of the language of mathematics in childrenÊs learning of mathematics demonstrated by the use of songs and nursery rhymes to teach young children in mathematics. Hence it identifies suitable words and terms to teach mathematical concepts and skills for various topics of early childhood mathematics. Lastly, it demonstrates the appropriate use of „math talk‰ to facilitate childrenÊs learning of mathematics. focuses on areas that pre-school children need to learn, such as, matching, classifying and ordering. It also discusses carrying out assessments on learning of matching, classifying and ordering. This topic also introduces children to the understanding of sets, and grouping of numbers and items in their categories. covers the relationship between patterns and algebra, while identifying the different types of patterns that pre-school children learn. Various activities are created within this topic for teaching and learning of algebra among preschoolers. discusses the understanding of number sense and how it is introduced to the pre-schoolers in the form of activities since children learn faster through play. Later the children are introduced to numbers using the concept of base-10 and its application. Understanding of place value helps the pupils to master multiplication, division, addition and subtraction. discusses the role of a teacher in facilitating problem solving learning amongst children. The teacher should be able to help pupils move their mind towards learning, using the concept of computation, and in using the easiest method of problem solving, and guiding them to use them freely and easily. gives an overview on what is meant by data, how to organise data and how to analyse data. This topic teaches children how to construct graphs using the data obtained. The concept of probability taught in this topic enables children to make logical thinking in their mathematical calculations introduces children in early childhood learning to the principle of measurement namely the standard and non-standard measurement. The formal approach and the techniques used to measure are also discussed. Through this teaching, the children are also exposed to the measurement of time; month, day, hour. covers methods in mathematics learning for early childhood education. It helps to identify the structure of social and emotional learning of pupils in

xii

 COURSE GUIDE

learning mathematics as well as identifying the instructional strategies to improve mathematics learning among children in the early childhood education. discusses the purpose of assessment and the basis and foundation of assessment in relation to pupilsÊ age groups. It also identifies the role of assessment in early childhood education. Besides knowing the different types of assessment strategies that could be used by teachers in assisting teaching methods, it is done in order to enhance the quality of pupilsÊ learning. Lastly this topic discusses the evaluation of assessment results.

TEXT ARRANGEMENT GUIDE Before you go through this module, it is important that you note the text arrangement. Understanding the text arrangement will help you to organise your study of this course in a more objective and effective way. Generally, the text arrangement for each topic is as follows: This section refers to what you should achieve after you have completely covered a topic. As you go through each topic, you should frequently refer to these learning outcomes. By doing this, you can continuously gauge your understanding of the topic. This component of the module is inserted at strategic locations throughout the module. It may be inserted after one sub-section or a few subsections. It usually comes in the form of a question. When you come across this component, try to reflect on what you have already learnt thus far. By attempting to answer the question, you should be able to gauge how well you have understood the sub-section(s). Most of the time, the answers to the questions can be found directly from the module itself. Like Self-Check, the Activity component is also placed at various locations or junctures throughout the module. This component may require you to solve questions, explore short case studies, or conduct an observation or research. It may even require you to evaluate a given scenario. When you come across an Activity, you should try to reflect on what you have gathered from the module and apply it to real situations. You should, at the same time, engage yourself in higher order where you might required to analyse, synthesise and evaluate insteadthinking of only having to recall andbedefine. You will find this component at the end of each topic. This component helps you to recap the whole topic. By going through the summary, you should be able to gauge your knowledge retention level. Should you find points in the

COURSE GUIDE

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xiii

summary that you do not fully understand, it would be a good idea for you to revisit the details in the module. This component can be found at the end of each topic. You should go through this component to remind yourself of important terms or jargon used throughout the module. Should you find terms here that you are not able to explain, you should look for the terms in the module. The References section is where a list of relevant and useful textbooks, journals, articles, electronic contents or sources can be found. The list can appear in a few locations such as in the Course Guide (at the References section), at the end of every topic or at the back of the module. You are encouraged to read or refer to the suggested sources to obtain the additional information needed and to enhance your overall understanding of the course.

PRIOR KNOWLEDGE No prior knowledge required.

ASSESSMENT METHOD Please refer to myVLE.

REFERENCES Bobis, J. Mulligan & Lowrie, T. (2009). Mathematics for children: Challenging children to think mathematically (3rd ed.). NSW. Pearson Education. Davis, G.A. and Keller, J.D. (2009). Exploring science and mathematics in a childÊs world (1st ed.). New Jersey: Pearson Education. Smith, S. S. (2009). Early childhood mathematics (4th ed.). Boston: Pearson. Charles, R. & Lind, K.K. (2007). Mathematics & science for young children (5th ed.). Clifton Park, NY: Thompson Delmar Learning Yelland, N. , Butler, D, & Diezmann, C. (1999). Early mathematical explorations. Needham Heights, MA: Pearson Publishing Solutions.

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TAN SRI DR ABDULLAH SANUSI (TSDAS) DIGITAL LIBRARY The TSDAS Digital Library has a wide range of print and online resources for the use of its learners. This comprehensive digital library, which is accessible through the OUM portal, provides access to more than 30 online databases comprising e-journals, e-theses, e-books and more. Examples of databases available Management are EBSCOhost, SpringerLink, Books24x7, Books, Emerald PlusProQuest, and Ebrary Electronic Books. As an InfoSci OUM learner, you are encouraged to make full use of the resources available through this library.

Topic  Introduction to

1

Early Childhood Mathematics

LEARNING OUTCOMES By the end of this topic, you should be able to: 1. Describe the various mathematics education;

misconceptions

about

early

childhood

2. Explain the theories of learning in relation to the learning of early childhood mathematics; 3. Suggest good practices that help to improve the quality of early mathematics education; and 4. Outline the content standards for early childhood mathematics.

 INTRODUCTION Early childhood mathematics education is an important aspect of early childhood education. It plays an important role in shaping childrenÊs future learning of mathematics. But ironically, unlike other levels of mathematics education, the development early and childhood mathematics and implementation of of teaching learning practices are stillcurriculum at the infancy stage.the It seems that there is inadequate effort to carry out proper control and monitoring of early childhood education at kindergartens in the country. Kindergartens are given the freedom to design their own early childhood mathematics programme. Mathematics at early childhood may appear to be very easy for teachers,

2

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TOPIC 1 INTRODUCTION TO EARLY CHILDHOOD MATHEMATICS

educators and even parents. But the teaching of early childhood mathematics should not be regarded as a simple task. It requires teachers who are well equipped with pedagogical content knowledge and skills, coupled with the right attitudes, in order to design effective learning activities for children.

1.1

MISCONCEPTIONS OF EARLY CHILDHOOD MATHEMATICS

There are many misconceptions about learning mathematics at early childhood. Such misconceptions may affect our understanding of how early childhood mathematics should be learned and taught. They may also become obstacles to the implementation of good practices in early childhood education. Some of these misconceptions are discussed below:

Mathematics is too tough a subject to learn for young children. Mathematics is often regarded as a formal subject that is difficult to teach as well as to learn. There are often doubts raised on whether mathematics should be taught and learned in a formal manner in early childhood education. Some people bring up the issue like „why should we be so cruel to involve young kids in such mind boggling maths activities?‰ and „They should be given more time to play and have fun, not learning subjects like mathematics.‰ Ironically, mathematics exists everywhere, including in childrenÊs everyday lives. They play and use mathematics without realising it. For example, when children arrange their toys or objects in the way they want, they are actually developing the concept of sorting and ordering in mathematics. When they search for missing parts of their toys, they are again involved in some form of counting.

In order to not burden the kids, we should only teach them basic numbers and basic shapes.

Teachers never underestimate the inherent mathematical and abilities ofshould children. In fact, there are growing evidences that showstrength that young kids enjoy solving mathematical problems through play. Early childhood mathematics can be both deep and broad. It should cover some major mathematics in a broad range of areas. These areas include numbers and

TOPIC 1

INTRODUCTION TO EARLY CHILDHOOD MATHEMATICS

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3

operations, geometry (shapes and space), algebra (patterns), measurements, problem-solving, and even the concept of statistics and data analysis.

In early childhood, the learning of language is more important than the learning of mathematics. Therefore our teaching should focus on language skills rather than mathematical skills. Both language and mathematics are important. In fact, education in mathematics is by itself part of the education on language and literacy. When we teach young children the language of mathematics, we are actually teaching them the skills of reading, speaking and writing mathematical language so that they are able to communicate mathematical ideas clearly. The learning of language enhances the learning of mathematics and similarly, acquiring logical thinking in mathematics also helps in enhancing language skills.

It is fine to teach a little mathematics, but it is irrelevant and unnecessary to carry out assessment on young children. It is necessary for teachers to assess childrenÊs mathematical knowledge and skills from time to time. This helps the teacher to understand the individual childÊs strengths and weaknesses and thus be able to identify appropriate remedial activities to help each of them improve. Mathematics evaluation should consider assessing children in multiple ways, not merely based on observations.

It is not suitable to teach young kids mathematics using the computer. Some teachers think that the computer is a bad learning tool for young children. They feel that the computer is an anti-social media. It is deemed as preventing children from actively interacting with one another. It is true that wrong use of the computer may hinder effective and healthy learning. But the problem does not lie with the computer. It is the teacher who should be responsible for the judicious use of the computer in helping children learn mathematics. The computer is inherently a very powerful learning tool, particularly for learning mathematics. If teachers are able to capitalise on the advantages of the computer and use the tool appropriately, childrenÊs learning of mathematics will certainly be enhanced.

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ACTIVITY 1.1 As a mathematics teacher, how would you educate the parents about the misconceptions of mathematics education for pre-school kids?

SELF-CHECK 1.1 Why are the teaching and learning of mathematics important at early childhood or pre-school level?

1.2 THEORIES IN EARLY CHILDHOOD LEARNING OF MATHEMATICS There are many educational and learning theories that focus on how children learn in the very early stages of life. These theories, which are collectively referred to as theories for early childhood development may be applied to many subject areas, including mathematics.

1.2.1

Piaget’s Theory of Child Development

Jean Piaget (1896-1980) was one of the child psychologists that had spent much time and effort studying the cognitive development of children. His theory has its basis in constructivism which emphasises that learning should be constructed from each individualÊs experiences. He also believed that that there is a need to establish the connection between previously learned concepts with newly acquired concepts for effective learning to take place. Piaget developed the famous . The theory basically focuses on the different stages of cognitive development an individual must go through in the growing-up process. Based on his theory, childrenÊs mental and cognitive development, starting from birth, can be divided into four different stages as illustrated in Figure 1.1.

TOPIC 1


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5

PiagetÊs Four-Stage Theory of Child Development

The four stages and their relationship to the learning of mathematics are briefly explained as follows: (a) This stage of cognitive development for children starts until they are about . The stage is characterised by childrenÊs progressive development of the concept of object permanence. Object permanence here means that children have the ability to look for objects that are out of their view. For example, a child is able to search and find his toy that may be placed in a drawer. At this stage, children rely very much on their five senses (smell, taste, hear, see and touch). They develop their understanding of ideas or concepts through concrete experiences. They are usually egocentric and are only able to see the world from their very own perspectives. (b) The development of the pre-operational stage takes place when children are . During this stage of development, the children are still mostly egocentric. They are also limited in their rational and logical thinking abilities. They are more comfortable working with concrete objects. They are now able to develop their language ability. They begin to understand the concept of sequencing and order and are able to solve one step logical problems using manipulative materials. It is possible at this stage to help children develop their basic skills in addition and subtraction. However, since they are still more comfortable with concrete objects, it is necessary to incorporate the use of concrete materials such as blocks and counters as tools for children to explore their basic mathematical ideas. Since children at this stage are quite confined to one-dimensional thinking, they are very much influenced by the visual representation of objects that they see.

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(c) This stage of cognitive development takes place when children are between . It is at this stage that children develop their logical thinking ability. They are able to classify and arrange objects based on specific features and characteristics rather than just focusing on visual representation. They are also able to view things from different dimensions and can accept that there may be multiple ways of getting an answer to a problem. (d) Children enter this stage of development at about and from there, continue their cognitive development till adulthood. At this stage, children do not rely solely on concrete experiences. They are now able to think in a logical and abstract manner. Therefore, it is now possible to teach them to analyse, synthesise and hypothesise. From the perspective of mathematics, children at the stage of formal operations are ready for abstract and higher order mathematical thinking. It is important for mathematics teachers of childhood education to understand the four stages of cognitive development. With such knowledge, a mathematics teacher would be able to determine the readiness of the children in learning a particular mathematically related skill. The teacher will also be able to plan learning activities which are appropriate for the children they teach. Since early childhood education is normally targeted at children at the pre-school level or an age range from 3 to 6 years old, it is thus more important for teachers of early childhood education to have a thorough understanding of the first two stages of child development, namely, the pre-operational stage and the concrete operational stage. Table 1.1 provides a summary of the characteristics of children at each stage of cognitive development and their implications to mathematics teaching and learning.

TOPIC 1




: PiagetÊs Four-Stages Theory of Cognitive Development and Mathematics Ability



egocentric;



learn using 5 senses;



rely on concrete;



limited use of words; and



(0 – 2 years )

understand object permanence.











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egocentric;



basic language skill;



learn using 5 senses; and





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rely on concrete experiences. 

(2 – 7 years)



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(7 – 12 years) 





(above 12 years) 

linking one object to another; linking numbers to objects; searching for hidden objects; counting concrete objects; and reciting numbers (such as 1 to 10). one-to-one correspondence; solve one-step mathematical problems using manipulative materials; classification, order and sequence; basic addition and subtraction; and differentiate objects by shapes and sizes.

logical thinking;



classifying based on features and characteristics; and



concept of dimensions.



basic functions; and



routine measurements.



abstract mathematical

reasoning, logical, abstract thinking; able to form hypothesis; and able to view things from various perspectives.



advanced seriation and classification; multiple solutions to one problem;

concepts; and comparing different mathematical situations.

7

8

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ACTIVITY 1.2 Present PiagetÊs Four Stages of Child Development and their implications to learning of mathematics in the form of a mind map.

1.2.2

Lev Vygotsky and the Social Development Theory

(1896-1934) was another famous child psychologist who is known for his . Unlike Piaget who emphasised that childrenÊs cognitive development is a pre-requisite for learning, VygotskyÊs theory focuses on the importance of social and cultural interaction on child development and learning. To Vygotsky, learning is mainly mediated by interactions. Interactions here mean social interactions among children and interactions between the learners and others who are „more knowledgeable‰. Vygotsky introduced the concept of . He argues that there is a gap between what a learner is capable of learning on his own and what the learner can achieve with support and help from peers, particularly those who are better (in knowledge) than them. Lev Vygotsky described this gap as the Zone of Proximal Development. According to Vygotsky, it is possible for children to expand their learning ability and achieve better understanding if they are allowed to learn in a meaningful learning environment with proper peer interactions, collaborations and guidance instead of just being taught by the teacher. This is what is referred to as crossing the ZPD (Figure 1.2).

: VygotskyÊs concept of ZPD

TOPIC 1


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9

Based on VygotskyÊs Theory, it is thus clear that a teacherÊs role is not just to teach. What is more important is that teachers should try to make the classroom to have a rich and an interactive learning community as much as possible. According to Piaget, children can construct and learn mathematics effectively by interacting with physical objects. While according to Vygotsky, learning of mathematics is more effective by giving children the opportunity to interact and work as a group. In other words, the teacher should always promote and encourage collaborative learning and group activities, not just the didactic form of classroom teaching and individualised learning.

1.2.3

Six Stage Theory of Zoltan P. Dienes

Zoltan P. Dienes (1916-2014) was a Hungarian mathematician whose name is synonymous with the Multi-base blocks (also known as Dienes blocks) which he invented for the teaching of place value. Dienes believed that to make learning of mathematics more appealing to young children, we should create an informal learning environment where children learn mathematics through playing games, singing songs, dancing, and using manipulative materials. Dienes is also famous for his . The six stages are: (a) At this stage children learn mathematics using a trial and error approach. They explore and discover by themselves mathematical ideas through interacting and experimenting freely with objects or situations. (b) This stage involves the use of games with rules. Games often come with rules, and mathematical ideas are often inherent in these rules. Therefore, the use of games helps children to develop the idea of rules in mathematics. (c) This involves guiding children to compare different games with similar rules but using different materials, or games with different rules. Such comparison helps children to understand the importance of rules and is also the first step towards introducing children to abstractions and how rules can be formulated. (d) At this stage, children are guided on diagrammatic representations of rules and abstractions. Proper use of arrow diagrams, pictures, charts, and tables help children to understand.

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(e) It is at this stage that children are introduced to the use of symbols, such as mathematical symbols, to represent the abstractions. (f) At this stage, children learn about formulating axioms, theorems, and writing proofs. Based on the Six Stage Theory, it is thus clear that initial learning of mathematics could be carried out in a not-so-formal manner, with activities and games that motivate children to play and learn.

1.2.4

Jerome Bruner’s Three Modes of Representation

Jerome Bruner (1915 - present) is a psychologist who has contributed a great deal to the theories of cognitivism and constructivism as well as theories related to childrenÊs cognitive development. To Bruner, teaching of mathematics should not be just focussed on memorising things like concepts, categories, and the traditional problem-solving procedures. Children should be allowed to explore, discover or construct their own knowledge. Bruner is well known for his proposed . He hypothesises that childrenÊs cognitive development can be represented by three different modes, namely, , and . Unlike PiagetÊs theory on stages of child development which emphasises the move of the children from one stage to another as they grow in age, Bruner views the modes of representation as ways in which children manipulate or organise their information. Children may progress from one mode to another as they grow, but the ability to operate in the higher mode does not mean the children will not think or operate in the other modes. (a) In the enactive stage, information on past events is stored in memory and presented in the form of motor responses. In other words, the ability to do something is displayed in the form of action such as doing a physical task, not by imagination or by describing with words. For example, a child might have learned from past experience that shaking a particular object such as a plastic bottle containing some sand would produce sound. He will then, from time to time, shake the bottle to demonstrate his ability to produce the sound. At this stage, children learn by doing, not by understanding.

TOPIC 1


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11

(b) At this stage, children visualise and store information in the form of images. We may describe this as having „pictures in the mind‰. ChildrenÊs thinking is associated with mental images or icons, and the mental images could be formed through seeing, hearing, smelling or touching the concrete objects. (c) Entering this stage, children begin to be able to store information in the form of symbols, or using languages. Languages are in fact powerful „symbols‰ that can be used to organise and describe information. Other symbols include numbers and music. The first two modes of representation, the enactive mode and the iconic mode are confined to concrete experiences with objects that children can see or touch. The symbolic mode deals with abstract concepts that may not be present physically but only exist as ideas and can be expressed using appropriate symbols such as language. It is possible to relate BrunerÊs theory to childrenÊs learning of early mathematics. At the early years of development, it may not be appropriate to teach children symbols. The enactive mode appears to be the pre-eminent mode for young children. Therefore, learning of mathematics should involve children physically performing some action. Some examples of action-based activities are as follows: (a)

Moving around;

(b)

Touching concrete objects that can be seen or touched;

(c)

Counting concrete objects that can be seen or touched;

(d)

Categorising objects according to colours; and

(e)

Categorising objects according to shapes.

When children are ready to operate at the iconic mode, the use of suitable pictures and images helps to enhance childrenÊs understanding of mathematics. Language and mathematics symbols play a very important role when children are able to demonstrate the symbolic mode of representation. It is at this stage that children begin to explore mathematical concepts which are somewhat abstract. From the above discussion, it is thus clear that for effective learning of early mathematics to take place, we must use suitable resources and employ the correct teaching and learning approaches that are aligned with childrenÊs predominant mode of representation. The summary of the theories related to Early Childhood Learning of Mathematics is illustrated in Table 1.2.

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Theories in Early Childhood Learning of Mathematics

SELF-CHECK 1.2 1. Why is the understanding of learning theories important for early childhood mathematics teachers? 2. What are the implications of VygotskyÊs Theory of Social Development on the teaching and learning of early childhood mathematics?

1.3

GOOD PRACTICES IN EARLY CHILDHOOD MATHEMATICS

As a teacher teaching mathematics at early childhood, it is necessary for us to know the aims and objectives of teaching early childhood mathematics so that we are able to plan our instructional activities that are geared towards achieving the intended aims and objectives. It is also necessary for us to study the various recommendations for childhood mathematics so that we are clear in our instructional strategies for giving quality mathematics education to our young children. We will see the recommendations of good practices for early childhood mathematics from other countries (USA and Australia) before going through our National Curriculum.

TOPIC 1

1.3.1

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13

NCTM Recommendations for Childhood Mathematics

The National Council of Teachers of Mathematics (NCTM) of the United States of America states that high quality, challenging and assessable mathematics education at the early childhood level (3 to 6 years) is an essential and important foundation for future learning of mathematics (NAEYC & NCTM, 2002). NCTM provides many recommendations to guide teachers and key professionals in classroom practices to achieve high quality mathematics education. Some of the key ideas extracted from these recommendations are that learning design and teaching practices of mathematics at early childhood should focus on the following aspects: (a)

Enhance childrenÊs interest and motivation in mathematics;

(b)

Build on childrenÊs pre-requisite knowledge and past experiences;

(c)

Focus on childrenÊs cognitive, linguistic, physical, and social-emotional development;

(d)

Ensure the coherence mathematical ideas;

(e)

Integrate mathematics with other activities and other activities with mathematics;

(f)

Provide sufficient time, learning materials and teacher support for children to engage in learning through play;

(g)

Employ varying strategies, methods and activities to promote mathematical ideas and concepts; and

(h)

Provide thoughtful and continuous assessments to support learning.

1.3.2

and

appropriate

sequencing

of

important

Recommendations from the Australian Association of Mathematics Teachers

The Australian Association of Mathematics Teachers published a position paper on early childhood mathematics in 2006 (Australian Association of Mathematics Teachers and Early Childhood Australia, 2006). The paper provides a number of recommendations for early childhood mathematics educators. Some of the recommendations which are beneficial to early childhood or pre-school mathematics teachers can be summarised as follows: (a)

Leverage on young childrenÊs curiosity to assist them in developing their mathematical ideas;

14

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(b)

Use suitable teaching approaches or curriculum that emphasise play, emergent numeracy, child-centred and child-initiated activities;

(c)

Ensure that mathematical ideas to be taught are relevant to childrenÊs present lives and also form a foundation to their future learning of mathematics;

(d)

Always motivate by recognising and celebrating childrenÊs achievements;

(e)

Encouragechallenging children tomathematical view themselves as mathematicians exploring problem-solving tasks; and be brave in

(f)

Design a learning environment that promotes social interactions among children as well as interactions with the teacher;

(g)

Provide suitable learning materials, space, time and other resource that encourage childrenÊs engagement in their mathematics learning;

(h)

Place emphasis on the proper use of the language to describe and explain mathematical ideas;

(i)

Recognise individual differences in terms of learning pace and ability and try to address the different learning needs;

(j)

Be sensitive to social-cultural differences among children when designing games or activities for children;

(k)

Encourage childrenideas; to participate actively in communicating and sharing their mathematical

(l)

Assess childrenÊs mathematical development using various means such as observation, discussion, listening to childrenÊs learning stories, interviewing children, and so on;


(m) Use assessment information not only to track childrenÊs development, but more importantly, to guide the teacher in planning follow-up interactions, tasks, activities and interventions which are more effective in terms of learning; and (n)

Always be prepared to spend time gathering new ideas and explore new pedagogical approaches to ensure children are provided with quality mathematics education.

The two sets of recommendations listed above serve as good guiding principles for teachers and educators who are involved in curriculum planning or teaching in early childhood mathematics education.

TOPIC 1

1.4


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15

CONTENT STANDARDS IN EARLY CHILDHOOD MATHEMATICS

Content standard refers to the set of mathematics contents that should be covered in the teaching and learning of early childhood mathematics. It represents the learning opportunities that should be provided to our young children in early childhood or pre-school mathematics. We will look at samples of our own local National Preschool Curriculum Standard (KSPK) and a few other countriesÊ proposed preschool curriculum standards.

1.4.1

Kurikulum Standard Prasekolah Kebangsaan (KSPK)

Currently in Malaysia, the Kurikulum Standard Prasekolah Kebangsaan (KSPK) or National Preschool Curriculum Standard provides initial experience in Mathematics to pre-schoolers (Table 1.3). KSPK briefly outlines what needs to be included in early mathematics: 

Pre-number concept;



Number;





Basic operations; Money;



Time; and



Shapes and space. Mathematics content in National Preschool Curriculum Standard (KPSK).

(ST 6.0) Pre-number experiences

(ST 6.1) Matching objects

(ST 6.1.1) Matching objects in pairs (ST 6.1) Matching not similar objects in pairs, such as cup and saucer. (ST 6.1.3) Matching two groups of objects: (a) Similar; and (b) Not similar (matching equal and unequal groups)

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(ST 6.2) Comparing quantity of objects

(ST 6.2.1 Comparing amounts to identify more, same or less)

(ST 6.3) Making seriation of one characteristics

(ST 6.3.1) Arranging objects according to criteria: (a) Small to big; (b) Short to long; (c) Low to high; and (d) Thin to thick and others.

(ST 7.0) Concept of Numbers

(ST 6.4) Identifying repetitive patterns and building patterns

(ST 6.4.1) Recognising repetitive pattern (ST 6.4.2) Copying repetitive pattern using objects

(ST 6.4.3) Recognising repetitive pattern (ST 6.4.4) Complete patterm given (ST 6.4.5) Creating repetitive pattern

(ST 7.1) Understanding numbers 1 to 10

(ST 7.1.1) Rote counting 1-10 (ST 7.1.2) Recognising

(ST 7.1.7) Ascending and descending numbers 1-10

symbols 1-10. (Example: 2-Two) (ST 7.1.3) Counting 1-10 (ST 7.1.4) Using dots to represent numbers. (ST 7.1.5) Matching symbols to quantity 1-10 (ST 7.1.6) Tracing numbers 1-10

(ST 7.1.8) Writing numbers 1-10

(ST 7.2) Recognising zero

(ST 7.3) Understand number 10-20

(ST 7.2.1) Writing zero (ST 7.2.2) Say zero (ST 7.2.3) Understand concept of zero (no value) (ST 7.3.1) Counting on from 1-20 (ST 7.3.2) Comparing 10 and 11 (11 is more than

TOPIC 1


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17

10) (ST 7.3.3) Write 1020 (ST 7.4) Understand sequence 20, 30, 40 and 50

(ST 7.4.1) Counting in 10Ês

(ST 8.0)

(ST 8.1) Understand

(ST 8.1.1) Able to

Mathematical operations

addition within 10operation

state the sum of 10 numbers within (ST 8.1.2) Counting all and counting on in addition (ST 8.1.3) Addition problem solving (ST 8.1.4) Writing mathematical statement (ST 8.1.5) Able to say out mathematical statement (ST 8.1.6) Solve number stories (ST 8.1.7) Tell situation in every life involving addition

(ST 8.2) Understand operation in substraction till 10

(ST 8.2.1) Removing objects from the group and able to count the balance (ST 8.2.2) Use everyday language to state the operation of subtraction (ST 8.2.3) Able to write mathematical statement (ST 8.2.4) Able to say mathematical statement (ST 8.2.5) Solve

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number stories (ST 8.2.6) Able to tell situation using subtraction (ST 9.0) Monetary value

(ST 9.1) Able to recognise different values of money

(ST 9.1.1) Able to recognise different values of money (ST 9.1.2) Able to arrange value of money in sequence (ST 9.1.3) Able to understand monetary exchange

(ST 10.0) Time concept

(ST 10.1) Understand time with reference to daily activities

(ST 10.1.1) Arrange events with time (ST 10.1.2) State the time for an activity

(ST 10.1.3) Able to state time (ST 10.1.4) Able to state days in a week (ST 10.1.5) Able to tell events (ST 10.1.6) Able to state months and days

(ST 11.0) Shape and space

(ST 11.1) Position of object in space

(ST 11.1.1) Able to state object in reference to environment (below, above...)

(ST 11.1.2) Able to recognise left and right (ST 11.1.3) Placing object in the right position

(ST 11.2) Able to recognise shape in the environment

(ST 11.2.1) Able to recognise circle, triangle and rectangle in the environment

(ST 11.2.2) Able to recognise square, triangle, rectangle and circle (ST 11.2.3) Able to construct cuboid from constructions like lego

Adapted from Kurikulum Standard Prasekolah Kebangsaan (BPK, 2010)

TOPIC 1

1.4.2


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19

Hawaii Preschool Mathematics Content Standard

The Hawaii Preschool Content Standard developed in 2004 by the School Readiness Task Force, Hawaii Good Beginnings Interdepartmental Council, described learning opportunities that should be provided for four-year-olds in early care and education programmes (Hawaii Good Beginnings Interdepartmental Council School Readiness Task Force, 2004). The standard is aimed to assist preschool teachers and administrators in their curriculum planning and lesson design. The mathematics content standard in particular, focuses on cognitive development of children through learning of mathematics. A summary of the mathematics content standard is provided in Table 1.4. : HawaiiÊs Pre-School Mathematics Content Standard

1. Numbers, numerals and basic numerical operations

2. Patterns and relationship between patterns



Recognising numbers



Counting numbers



Recognising names and symbols of numbers



Count up to 10 objects in meaningful context



Addition and subtraction using fingers or





Sorting and classifying objects by colour



Sorting and classifying objects by function









3. Shapes and Space concepts

objects Sorting and classifying objects by size, number and other properties in meaningful context

Sorting and classifying objects by colour and material Order objects by properties (small to large, light to heavy, and so on) Recognising patterns Creating patterns using manipulatives in selfdirected activities





Recognising 2D and 3D shapes Naming 2D and 3D shapes



Creating 2D and 3D shapes



Comparing 2D and 3D shapes



Recognising geometric shapes in daily lives

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4.

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Measurement concepts and use of measurement



Creating art with shapes and symmetry



Demonstrating spatial understanding in play







5.

Representing and interpreting data



Understanding measurement attributes (length, volume, weight, area, time) Comparing attributes in meaningful context Measuring using standard and non-standard units Gathering information about themselves and about surroundings



Contributing data for simple graphs



Interpreting and discussing based on graphs

: Hawaii Good Beginnings Interdepartmental Council School Readiness Task Force (2004)

1.4.3

NCTM’s and NAEYC’s Proposed Preschool Mathematics Content Standard

The National Association for the Education of Young Children (NAEYC) and the National Council of Teachers of Mathematics (NCTM) of The United States of America put forward on mathematics education in 2002, and updated it later ina joint 2010 position (NAEYCpaper & NCTM, 2002). The paper outlines the learning path and the teaching strategies that may be used for teaching mathematics to young children between 3 to 6 years old. The information serves well as a standard for early childhood mathematics education. A summary of the learning path and the proposed strategies is provided in Table 1.5. Early Childhood Mathematics Learning Content and Strategies proposed by NAEYC & NCTM

1. Numbers and operations



Counting numbers



Linking items with numbers







Counting 1, 2, 3, ⁄10 Counting groups of 10 up to 100 (10, 20, 30... 100) Non-verbal addition and subtraction with small numbers



Addition using counting strategy



Subtraction

TOPIC 1

2. Geometry sense


and

spatial















3. Measurement

4. Pattern/algebraic thinking

Sorting and classifying objects by colour Sorting and classifying objects by function Sorting and classifying objects by colour and material Order objects by properties (small to large, light to heavy, and so on) Recognising patterns Creating patterns using manipulatives in selfdirected activities Recognising 2D and 3D shapes



Naming 2D and 3D shapes



Creating 2D and 3D shapes



Comparing 2D and 3D shapes



Recognising geometric shapes in daily lives



Creating art with shapes and symmetry



Demonstrating spatial understanding in play







21

Sorting and classifying objects by size, number and other properties in meaningful context





5. Displaying and analysing data



Understanding measurement attributes (length, volume, weight, area, time) Comparing attributes in meaningful context Measuring using standard and non-standard units Gathering information about themselves and about surroundings



Contributing data for simple graphs



Interpreting and discussing based on graphs

NAEYC & NCTM (2002). Early Childhood Mathematics: Promoting Good Beginnings

SELF-CHECK 1.3 1. What do we mean by content standard? 2. How are the early childhood mathematics content standards adopted by other countries useful to us?

22

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










Early childhood mathematics education is an important aspect of early childhood education. Misconceptions in early childhood mathematics may be an obstacle to effective planning and implementation of early childhood mathematics. Thorough understanding of learning theories and theories of childhood development helps in the teaching of early childhood mathematics. It is beneficial to study recommendations and good practices in the teaching of childhood mathematics. Early childhood mathematics content standards adopted in other countries serve as a benchmark for us to design our own early childhood mathematics content.

Content Standard

Six Stage Theory

Early Childhood

Social Development Theory

Good Practices

Theories of Learning

Interactions

Zone of Proximal Development

Misconceptions

Australian Association of Mathematics Teachers and Early Childhood Australia (2006). Early Childhood Mathematics. Retrieved from http://www.aamt.edu. au/content/download/722/19512/file/earlymaths.pdf Bahagian Pembangunan Kurikulum (2010). Kurikulum Prasekolah Kebangsaan: Kurikulum Standard Prasekolah Kebangsaan . Retrieved from http://web.moe.gov.my/bpk/v2/index.php?option=com_content&view=art icle&id=310&Itemid=473&lang=en

TOPIC 1


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23

Hawaii Interdepartmental Council on Children and Families & School Readiness Task Force (2004). Hawaii preschool content standards: Curriculum guidelines for programs for four-year-olds. Retrieved from http://www. goodbeginnings.org/pdf/SchoolReadiness/HPCS2004.pdf NAEYC & NCTM (2002). Early Childhood Mathematics: Promoting Good Beginnings. Retrieved from http://www.naeyc.org/files/naeyc/file/ positions/psmath.pdf

Topic  The Language of Mathematics

2

LEARNING OUTCOMES By the end of this topic, you should be able to: 1. Explain the role of language and the language of mathematics in childrenÊs learning of mathematics; 2. Demonstrate the use of songs and nursery rhymes to teach young children mathematical language; 3. Identify and use suitable words and terms to teach mathematical concepts and skills for various topics of early childhood mathematics; and 4. Give examples of the appropriate use of mathematical talk to facilitate childrenÊs learning of mathematics. 5. Justify the importance of integrating mathematics into childrenÊs play.

 INTRODUCTION AdultsÊ thinking about mathematics is very different from children. To most adults, mathematics is about numbers, formulas and calculations. But to children, mathematics is something (concepts and skills) that allows them to process and to establish the connections between the different elements or objects that they perceive with their senses. The use of mathematical concepts and skills such as comparing, measuring, patterning and ordering helps children to understand phenomena, appreciate design, and solve problems they face and to make predictions. ChildrenÊs initial experience with mathematics is basically intuitive. For example, a child experiences the concept of „small‰ and „big‰ when he is „moved‰ from

TOPIC 2

THE LANGUAGE OF MATHEMATICS



25

his cot to his parentsÊ bed next to the cot. As children grow, they gradually move from intuitive thinking to a more organised form of mathematical thinking. Since mathematics is about thinking, not just about doing, it is important to stress that leaving children alone playing and exploring with manipulatives such as blocks or peg boards is not sufficient to develop childrenÊs formal mathematical thinking. To facilitate the move from intuitive thinking to formal mathematical thinking, parents and teachers need to provide children with appropriate vocabulary and dialogues that promote childrenÊs understanding of and reflection on mathematics. Thus language plays an important role in childrenÊs development of mathematical concepts and skills.

2.1

THE ROLE OF LANGUAGE AND LANGUAGE OF MATHEMATICS

Language is a means of communication. We use language to communicate our ideas and thoughts to other people. Likewise, the language of mathematics is also a means of communication, albeit means to communicate mathematical ideas and thought. The language of mathematics is however inseparable from the language itself. This is because whenever we want to express mathematical ideas, the language of mathematics we use is always part of the language we use for our verbal or written communication. Babies and children rely on their five senses to learn about the world around them. As they grow, they gradually acquire the concepts about language through interactions with people around them; and at the same, they also gradually form mathematical ideas and concepts long before they understand what the four arithmetical operations are (addition, subtraction, multiplication and division). The ability of children to communicate basic mathematical ideas and concepts using proper mathematical language provides the strong basis for future learning of mathematics. Children learn best when they are at their natural setting. This is true for both the learning of a language or for the learning of mathematics per se. Children in early childhood are exposed to a world full of mathematical experiences. They are often involved in activities like , , and as they play and interact. Throughout childrenÊs play, language is used as a means for children to express their mathematical thinking. It is therefore important for the teacher to be able to „teach‰ at the right moment using the right language so as to help children build the connection between daily experiences with mathematical concepts and ideas. Appropriate use of language enhances childrenÊs mathematical

26

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TOPIC 2 THE LANGUAGE OF MATHEMATICS

literacy. Similarly, well designed mathematical learning activities, by way of childrenÊs plays, help to enhance language literacy. There are various strategies that can be used to develop childrenÊs mathematical language skills. These include activities like drama, childrenÊs literature, role play, story-telling, singing songs or rhymes, and many others that you can think of.

2.2

DEVELOPING LANGUAGE USING SONGSMATHEMATICAL AND NURSERY RHYMES

Music, songs and rhymes are useful teaching media for young children. Effective use of songs or rhymes motivates and enhances childrenÊs learning of mathematics. There are many songs and nursery rhymes that are specifically written to help teachers teach mathematical language as well as mathematical concepts and skills to young children. In this kindergarten lesson, pupils may use the nursery rhyme "Baa Baa Black Sheep" to practice non-locomotor skills by acting out the rhyme, or develop number sense by participating in a number-matching activity. Math nursery rhymes are an effective way for introducing mathematical ideas and concepts simply because most young rhymes childrenand enjoy listening to musicare or songs with nice rhythm. Many nursery songs for children designed to guide childrenÊs learning in a step-by-step manner. Using songs to learn mathematics makes learning more fun and interesting. Learning is not deliberately taught but comes by naturally as children enjoy singing the nursery rhymes and songs that teach mathematical ideas and concepts. In this way, it is easier for children to remember numbers and how to count numbers. Very often, the nursery rhymes or songs demonstrate how mathematics can be applied in childrenÊs daily activities and in this way, mathematics becomes less abstract but more meaningful and more interesting to the young children. The following tables show three popular songs or nursery rhymes and the mathematical language, concepts and skills that can be taught using these songs or rhymes:

TOPIC 2




(a) Baa Baa Black Sheep Nursery Rhyme and its Relation to Learning of Mathematics for Young Children Song/Rhyme Baa, baa black sheep have you any wool? Yes sir, yes sir three bags full One for my master one for my dame One for the little boy Who lives down the lane Mathematical Concepts or Ideas

Mathematical Language

URL to listen to the song

(b)



Number sense



Sharing, dividing



Counting



Have you any



Three bags full



One for ⁄



Down the lane



http://www.youtube.com/watch?v=DBnLuOmYML0



http://www.youtube.com/watch?v=gBEHFFnV3RY

This Old Man Nursery Rhyme and its Relation to Learning of Mathematics for Young Children

Song/Rhyme This old man, he played one, He played knick-knack on my thumb; Knick-knack paddywhack, Give a dog a bone, This old man came rolling home. This old man, he played two, He played knick-knack on my shoe; Knick-knack paddywhack, Give a dog a bone, This old man came rolling home. This old man, he played three, He played knick-knack on my knee; Knick-knack paddywhack, Give a dog a bone, This old man came rolling home.

27

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This old man, he played four, He played knick-knack on my door; Knick-knack paddywhack, Give a dog a bone, This old man came rolling home. This old man, he played five, He played knick-knack on my hive; Knick-knack paddywhack, Give a dog a bone, This old man came rolling home. This old man, he played six, He played knick-knack on my sticks; Knick-knack paddywhack, Give a dog a bone, This old man came rolling home. This old man, he played seven, He played knick-knack up in Heaven; Knick-knack paddywhack, Give a dog a bone, This old man came rolling home. This old man, he played eight, He played knick-knack on my gate; Knick-knack paddywhack, Give a dog a bone, This old man came rolling home. This old man, he played nine, He played knick-knack on my spine; Knick-knack paddywhack, Give a dog a bone, This old man came rolling home. This old man, he played ten, He played knick-knack once again; Knick-knack paddywhack, Give a dog a bone, This old man came rolling home. Mathematical Concepts or Ideas



Numbers one to ten



Counting 1 to 10



Relate words (one, two, three, ..., ten) to symbols (1, 2, 3,⁄, 10)

TOPIC 2

Mathematical Language URL to listen to the song




Digits 1 to 10



Give a dog a bone



On my (thumb, shoe, door and so on)



http://www.youtube.com/watch?v=tYAI2b2P30Y



http://www.youtube.com/watch?v=CmPDYkCBuoY



(c) Hickory Dickory Dock Nursery Rhyme and its Relation to Learning of Mathematics for Young Children Song/Rhyme Hickory Dickory dock, The mouse ran up the clock, The clock struck one The mouse ran down, Hickory Dickory dock. Hickory Dickory dock, The mouse ran up the clock, The clock struck two And down he flew, Hickory Dickory dock. Hickory Dickory dock, The mouse ran up the clock, The clock struck three And he did flee, Hickory Dickory dock. Hickory Dickory dock, The mouse ran up the clock, The clock struck four, He hit the floor, Hickory Dickory dock. Hickory Dickory dock, The mouse ran up the clock, The The clock mousestruck took afive, dive, Hickory Dickory dock. Hickory Dickory dock,

The mouse ran up the clock, The clock struck six,

29

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That mouse, he split, Hickory Dickory dock. Hickory Dickory dock, The mouse ran up the clock, The clock struck seven, 8, 9, 10, 11, Hickory Dickory dock. Hickory Dickory dock, The mouse ran up the clock, As twelve bells rang, The mousie sprang, Hickory Dickory dock. Hickory Dickory dock, "Why scamper?" asked the clock, "You scare me so I have to go! Hickory Dickory dock." Mathematical Concepts or Idea

Mathematical Language

URL to listen to the song



Telling Time with a clock



Counting 1 to 12



Relate words (one, two, three, ..., ten) to symbols (1, 2, 3,⁄, 10)



The clock struck one, two ⁄



As twelve bells rang



Up



Down



http://www.youtube.com/watch?v=MTBRxmvPGus



http://www.youtube.com/watch?v=Kx3W8qo08XM

TOPIC 2


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31

ACTIVITY 2.1 1. Search on the Web for childrenÊs songs or nursery rhymes that can be used to teach mathematical ideas or concepts related to: (a)

Shapes; and

(b)

Measurement.

2. Discuss how would you use songs to teach the related mathematical concepts with your coursemates.

2.3

USEFUL WORDS TO GUIDE LEARNING OF EARLY MATHEMATICAL CONCEPTS

It is beneficial for early childhood mathematics teachers to maintain a vocabulary of words that constitute an important component of mathematical language. Teachers who are familiar with the use of such words will be able to communicate clearly their mathematical ideas to the young children. The following table gathers some of the useful words commonly used when teaching mathematical language and skills to young children. Useful Words in Mathematical Language

Number Sense and Concept of Quantity

big, small, bigger, smaller, biggest, smallest, many, too many, how many, few, fewer, too few, enough, not enough, none, some more, count, different, more, less, the same, less than, greater than

Four Operations (+,–, ×, ÷)

sum, total, add, minus, take away, plus, times, multiply, how many times, how many more, how many left, share equally, divide

Patterns and Relationship between Patterns

before, after, next, first, second, third, last, above, below, between, biggest, match, list, pattern, smallest, bigger, smaller, one more, ten

Shapes and Space

more, one less, ten less, arrange, same, different round, sides, corners, flat, box, carton, circle, square, triangle, tube, stairs, room, container, more, less, more than, less than

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Measurement of Length and Distance

long, short, longer, shorter, longer than, shorter than, longest, shortest, height, width, high, low, deep, shallow, thick, thin, tallest, , far, near

Measurement of Weight

heavy, light, heavier, lighter, heaviest, lightest, balance, weigh, weigh more, weigh less

Measurement of Time

morning, afternoon, evening, day, night, today, tomorrow, hours, minutes, seconds, long hand, short hand, fast, slow, faster, slower, fastest, slowest, long, short, longer, shorter, longest, shortest, now, soon, early, late, old, older, oldest, new, newer. newest, takes longer time, take less time, clock, before, after

Measurement of Money

money, coin, price, how much, buy, sell, spend, pay, change, cheap, cheaper than

Words Used to Make Comparison

high – low big – small large – small more – less loud – soft young – old tall – short heavy – light

2.4

MATHEMATICAL TALK AND EARLY MATHEMATICAL CONCEPTS

From as early as the first year of life, children gradually develop their understanding of mathematical concepts through everyday experiences, routine activities and their interactions with people around them, particularly their trusted parents. Very often, children acquire the understanding of mathematical concepts or ideas in an informal way, such as through play. Unfortunately, many adults do not realise the importance of play in the learning of mathematics. As a result, they miss the opportunity to help children enhance their understanding of mathematics during play by using the right „maths talk‰ at the right moment. Another important aspect is the right use of language. This is because if we want to „talk maths‰ effectively to the children, we have to communicate with them using the language that they understand.

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33

Mathematics is actually part of our everyday lives. In fact, mathematics is everywhere. We apply mathematics and use the mathematical language from time to time without realising it. For example, when we iron our clothes and then arrange the clothes into different categories, we are actually using the mathematical skills of sorting and classifying. When we teach young kids to sing the ABC rhymes, we are instilling the concept of ordering (A, B, C, ⁄). Generally, there are six categories of „maths talk‰ of which children are involved in during their free play activities. These six categories are: (a)

Classification;

(b)

Magnitude/size;

(c)

Enumeration;

(d)

Patterns and shapes;

(e)

Spatial relations; and

(f)

Concepts of part and whole.

(a) Classification refers to the arrangement of objects into different groups according to their criteria. For example, children may sort wooden blocks into groups according to their colours or shapes.

Classifying according to colours

34

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(b) Magnitude refers to making evaluation of items or making comparison between items based on the magnitude. For example, a child may compare two toys and claim that his toy is bigger than his friendÊs toy.

Comparison between items to show difference in magnitude

Some of the common terms in the mathematical language used by the teachers to help to understand the concept of magnitude are as follows: 

long, longer, longest



short, shorter, shortest



fast, faster, fastest



more, less



big, bigger, biggest



small, smaller, smallest



large, larger, largest



some, a lot, little

Can you think of other terms that early childhood mathematics teachers can use to convey the idea of magnitude to children? (c) Enumeration here means making numerical judgment or quantification. For example a child may be taught to say he is 5 years old. Another child may express that he has more pencils than his friend who sits next to him by saying „I have two pencils. You have one!‰

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35

Some of the common terms in the mathematical language used by the teachers to help to understand the concept of enumeration are as follows:

Comparison between items to show difference in magnitude

(d) This involves the exploration of patterns and spatial forms. For example, children may learn to arrange objects in an orderly manner following a certain regular pattern. They may also be involved in completing a puzzle by selecting pieces that match certain shapes.

Children may learn to arrange objects according to a certain pattern

Some of the common terms in the mathematical language used by the teachers to help to understand the concept of patterns and shapes are as follows:

square, rectangle, triangle, circle, cube, line, box, round

36

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(e) This involves exploration of relative positions, directions and distances between objects. For example, a child may describe how the different Lego bricks should be positioned to form an object by saying that „this one should be „on top‰ of that one‰.

Activity involving spatial relations includes playing with Lego bricks and articulating Âmaths talkÊ.

(f) The concept of part and whole can be introduced to young children through free play activities that involve sharing of toys or food. Mathematics teachers may use the terms like, ÂpartÊ, ÂwholeÊ and ÂhalfÊ when asking children to give some of what they have to their friends. For example, the teacher may direct a child to „give half of the cake‰ to his partner in a free play activity.

2.4.1

Examples of Mathematical Talk

The following Table 2.5 lists some examples of mathematical concepts that can be introduced to young children via informal but intentional mathematical mediated dialogues or in short, „maths talk‰. Common Intentional Mathematical Mediated Dialogue

1. Number Sense

Counting 1, 2, 3, ⁄ Asking „how many ⁄?‰ Asking „which is bigger?‰ Asking „which is more?‰

TOPIC 2


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2. Number Operations

Asking „how many all together?‰ Asking „how many left?‰ „Sharing‰ the toys Asking „How many do you get?‰

3. Patterns and Relationships between Patterns

Arrange „big-small-big- small‰ Arrange from the smallest to the biggest Move „up‰ ,„down, „up‰, „down‰

4. Shapes and Space Concepts

Name the shapes (2D and 3D shapes) Ask question like ‰give me a triangle/square/ sphere‰

5. Measurement Concepts

Compare the weight/size/length Ask „which is heavier?‰ Ask „which is bigger?‰ Ask „which is longer?‰ Ask „which is the heaviest?‰ Ask „which is the biggest?‰ Ask „which is the longest?‰

6. Data Display and Analysis

Group similar things together Count how many in each groups

37

Draw pictures to show

Initiating mathematical talk judiciously and at an appropriate time help children not only to acquire the proficiency in the language of mathematics, but also to achieve a better understanding of the mathematical ideas and concepts inherent in the language used. It needs to be stressed that even though children normally learn mathematics in an unintentional manner, through activities that are enjoyable to them, such learning activities need to be carefully planned by teachers so as to achieve desirable learning outcomes.

2.5

INTEGRATING MATHEMATICS INTO CHILDREN’S PLAY

Since play is part on andplay parcel of childrenÊs mathematics teachers should smartly leverage activities as theworld, gateway to engaging children in meaningful mathematical inquiry and learning. Mathematics learning experiences for children can be acquired using two main approaches:

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(a)

By

(b)

By


; and that involve some form of mathematics.

In this respect, childrenÊs learning of mathematical concepts and skills is not deliberately carried out and may not appear obvious to the children. Since play is childrenÊs core activity, it would be better if teachers could design play activities integrate theapproach play environment. In other words,that carrying out mathematics play activitiesseamlessly may be a into better for young children. It needs, however, adequate knowledge and skills for teachers to be able to create learning environment with appropriate tasks, challenges and language to support effective understanding of mathematics. It also needs to be noted that whatever the approach used, childrenÊs play activities must be designed to suit the ages of the children involved.

ACTIVITY 2.2 Design a play activity that involves children playing directly with the following mathematical concepts: (a)

Magnitude; and

(b)

Spatial Relations.

SELF-CHECK 2.1 1. Why is the use of mathematical language important for childrenÊs development of mathematical thinking? 2. Why is it that integrating mathematical concepts into childrenÊs play is more effective than teaching children mathematics as a subject?

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Developing childrenÊs thinking should begin from early childhood. The use of manipulatives alone is inadequate in developing childrenÊs mathematical thinking. Teachers should use appropriate vocabulary and dialogue to facilitate the developing of childrenÊs mathematical thinking.

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Both language and mathematical language play important roles in childrenÊs development of mathematical concepts and skills. Teachers should identify useful words or terms that help children understand mathematical concepts of various topics ChildrenÊs world is associated with play. Therefore, it is a better approach to integrate mathematical concepts and skills seamlessly into childrenÊs daily play activities as compared to teaching mathematics in a direct manner.

Free play

Mathematical talk

Integrate mathematics into play

Mathematical vocabulary and dialogue

Language

Six categories of mathematical talk

Mathematics Mathematical language Mathematics Mathematical songs and rhymes

Topic  Early

3

Mathematical Concepts: Classifying, Matching, Sorting and Ordering

LEARNING OUTCOMES By the end of this topic, you should be able to: 1. Identify areas that pre-school children need to learn about matching, classifying, matching, sorting and ordering; 2. Design instructions for the learning of matching, classifying, matching, sorting and ordering; and 3. Carry out assessments on matching, classifying, matching, sorting and ordering.

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INTRODUCTION

Children build their conceptual understanding of mathematics in a sequential manner with one concept built upon another. Initial learning is normally

TOPIC 3

EARLY MATHEMATICS CONCEPTS: CLASSIFYING, MATCHING, SORTING AND ORDERING

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41

acquired through the process of childrenÊs free exploration and concrete experiences with materials that they come across in their daily activities. One of the earliest mathematical concepts that children develop is the concept of . The attributes that distinguish one material from another include colour, size, shape, height, width, length, and weight. The understanding of the concept of attributes forms the basis for the learning of many other higher mathematical concepts. Piaget relates the learning of physical knowledge about colour, size, shape and texture to childrenÊs construction of logico-mathematical knowledge which is the basic foundation of mathematical thinking. The understanding of attributes allows children to observe the differences between objects. It is with such understanding that they are then able to perform the tasks of , , and . The ability to carry out such tasks is an important foundation for future learning of algebraic concepts. Besides that, the learning activities related to classifying, matching, sorting, categorising and ordering also give children the opportunities to develop their ability to reason, solve problems, make decisions and to function as an independent learner.

3.1

CLASSIFICATION

One of the early mathematical concepts that children acquire through free or guided exploration is the skill of classification. To perform the task of classifying objects, children need to be able to discriminate, match, sort, group or categorise objects based on their attributes. The knowledge about classification is the basic foundation to understanding the concept of sets in mathematics. Children need to use their senses to observe the various sets of objects, and also to identify the similarities and differences between these objects. Classification requires children to group objects with one or more common features or attributes. The following Table 3.1 shows a list of attributes and the attributed values at the qualitative level. List of Attributes and their Qualitative Values

Colour

green, red, blue, yellow, black, white

Size

big, small, large, little

Shape

square, rectangle, circle, triangle

Height

tall, short

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Length

long, short

Width

wide, narrow, thick, thin

Weight

heavy, light

In the process of developing the concepts and skills of classification, a child needs to go through the following logical or Piagetian sequence of learning: (a)

Learning to discriminate, that is, to decide whether the two items given to him are the same or different;

(b)

Learning to match and group items based on a specific attribute; and

(c)

Learning to classify or categorise items based on a combination of two or more attributes.

In view of the importance in establishing the logical sequence of conceptual development, the teacher should design or create learning environments that promote learning in the sequential manner. Figure 3.1 shows the suggested sequence and serves as a guide for the teacher to plan the learning activities for the young children:

: Suggested sequence of activity to guide teacher in planning mathematics learning activities for young children

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The Understanding of Sets

A set can be defined as a collection of things or objects with some pre-defined common characteristics. It is important to define the common characteristics or criteria clearly so as to be able to determine which objects are to be included in the set. For example, a set of „animals‰ is different from a set of „animals with four legs‰. A Venn diagram can be used to group objects into sets and subsets (Figure 3.2), as well as to describe the intersections and unions of sets.

A Venn diagram with subsets

Given two sets A and B, the intersection is the set that contains elements or objects that belong both to A and to B. For example, set A is the set that contains all pupils in class Alpha who bring food to school, and set B is the set that contains all ofpupils Alpha who bring school. Then food the intersection A andinB class contains pupils from classmoney Alpha to who bring both and money to school. Figure 3.3 illustrates the meaning of intersection of sets.

A Venn diagram showing intersection of set A and set B

A common mistake made by children in classifying objects or items into different sets is that they tend to assume that a set can only contain objects which are identical. In actual fact, objects in a set may not necessarily be identical. The inclusion of the objects in a particular set depends on the definition of the set. Therefore, it is important for teachers to make clear to pupils the criteria that define the set before asking them to carry the task of classifying objects into different sets.

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ACTIVITY 3.1 Collect pictures of 5 different childrenÊs toys that can be used to help children learn classification based on attributes. Write down clearly what kind of learning can be acquired for each toy you identify.

SELF-CHECK 3.1 What do you understand by union of 2 sets? Can you explain the meaning of union of sets?

3.2

MATCHING

Matching is the concept of one-to-one correspondence. It is an important concept because it forms the basis for understanding our number system, noting that we give meaning to each number (1, 2, 3, ⁄) by matching the number to the quantity of items represented by that number. Table 3.2 shows some examples of the activities that can be used to help children learn about matching. Samples of Learning Activities to Help Children Learn about Matching

Children are asked to place each of the coloured cards provided into the right hole on the wooden board 1. Identifying shapes that are similar and shapes that are different 2. Matching shapes which are identical

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1. One-to-one correspondence 2. Geometrical shapes

Children are asked to move the rectangular strips provided to the right position of the drawing 1. 2. 3. 4.

Identifying identical shapes based on length and width Matching shapes of the same colour Matching shapes which are identical Recognising shapes and colours

1. One-to-one correspondence 2. Geometrical shapes

3. Length and width

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3.2.1


Challenging Matching Task

The matching task becomes more challenging to children when: (a)

The items to be matched are identical but they are not symmetrical;

(b)

There are too many items to be matched;

(c)

The two sets to be matched do not have equal number of items; and

(d)

The two items or objects to be matched are not identical.

The following diagrams illustrate each of the above cases.

: The items to be matched are identical but not symmetrical

There are too many items to be matched

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The two sets to be matched do not have equal number of items

The two items or objects to be matched are not identical. Children need to understand the relationship between the two items to be matched

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ACTIVITY 3.2 Describe how you would measure the different levels of achievement of children in performing a matching task. Write your answer in the myVLE forum.

3.3

SORTING, CATEGORISING AND GROUPING

Children who have developed the skills of sorting, categorising and grouping should be able to sort out things or items which are alike, from among a mixed group of items. They should also be able to group items based on certain criteria. The idea of sorting, categorising and grouping are important mathematical ideas that lead to the concept of sets. As such, understanding and acquiring these ideas and skills will be beneficial to children when they study about sets in their future learning of algebra. Table 3.3 shows some examples of the activities that can be used to help children learn about sorting, categorising and grouping. Example of Activities Involving Sorting, Categorising and Grouping

Children are asked to sort out the buttons according to colours and then group buttons of the same colours together 1. Identifying colours that are different 2. Sorting out things of the same colours

1. Concept of sets 2. Organising data and displaying data

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ORDERING OR SERIATION

Ordering involves arranging three or more objects into a particular sequence based on certain pattern of arrangement. Some of the factors that determine the pattern of arrangement include (from big to small, or vice versa), (from long to short, or vice versa), (from tall to short, or vice versa), (from light to dark, or vice versa) and (from more to less, or vice versa). Since ordering involves arranging three or more objects, it is a more difficult task than just making a comparison between two objects. A very young child below two years old might not be able to seriate or arrange objects at all. Normally, a two to four year old child who might not be able to see the pattern of order could make some effort to arrange the objects in a random manner. But a five year old child might be able to arrange objects in a certain pattern or order through trial and error. Children of six years or above could possibly be able to arrange objects through a systematic way. Such changing ability of a typical child is in alignment with the stages of cognitive development put forward by Piaget. Ordering is an important concept to learn. It forms the basis of our numbering system, noting that in our number system, the numbers are arranged in order of hierarchy from small to big (1, 2, 3, 4, 6, ⁄). Furthermore, well-developed seriation skills also help to develop childrenÊs higher order thinking skills and problem-solving skills. Activities designed for learning about ordering or seriation can range from basic seriation involving ordering one set of objects, to double seriation involving ordering two sets of objects, or reverse seriation which involves reversing the order of arrangement. Seriation skills can be taught and reinforced using suitable developmentally appropriate hands-on activities. Some activities that help children acquire the concept and skills of ordering or seriation are shown in Tables 3.4 and 3.5:

SELF-CHECK 3.2 How do PiagetÊs stages of cognitive development relate to planning of activities involving matching, sets and classification, and order or seriation?

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: Activity Involving Single Seriation

1. 2.

Children are given a number of hollow cylinders (say, 4) with same heights but different diameters Children are asked to place the smaller cylinder into the bigger one in the correct order until all cylinders are inside the biggest one

1. 2. 3.

Identifying objects of different sizes Differentiating and selecting objects according to size Recognise the order of the sizes

1. 2.

Concept of size Concept of order of size : Activity Involving Double Seriation

1. 2.

3.

4.

Children are given three similar boxes but with different sizes By referring to the boxes, the teacher explains and describes to children the meaning of big, medium-size and small The teacher asks the children to show the boxes according to the size specified by the teacher (big, medium-size, or small) The teacher asks the children to place the boxes one top of the another in order of size, from big to small

TOPIC 3

3.5


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5.

The teacher asks the children to place the boxes one top of the another in order of size, from small to big (concept of reverse seriation)

1. 2.

Identifying objects of different sizes Differentiating and selecting objects according to size of „big‰, „medium-size‰ and small

1. 2. 3.

Concept of size Concept of order of size Concept of reverse seriation

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INFORMAL LEARNING AT HOME OR AT SCHOOL

Informal learning is always an important part of preschool childrenÊs education. Unlike formal learning which is very structured and has clearly defined learning outcomes to be achieved, informal is never organised, and has no pre-set objectives. Learning acquired through informal learning appears to be unintentional. It is sometimes described as learning by experience. Nevertheless, it is still possible to create informal learning environments that allow children experience mathematics that they need to learn. Parents or teachers can help in creating such an environment that promotes the learning of classifying, matching, sorting and ordering by doing the following: (a)

Make sorting part of the daily routines;

(b)

Involve children in games that require them to perform matching tasks;

(c) (d)

Offer activities that involve classifying; and Encourage children to observe similarities and differences between objects that they come across and be able to make comparisons.

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3.6


ASSESSMENT ON CLASSIFYING, MATCHING, SORTING AND ORDERING

ChildrenÊs development of mathematical concepts is sequential. Therefore, learning activities for children should be planned in such a way that the activities provide developmentally appropriate learning experiences for them. In order to ensure that the activities are developmentally appropriate, there is a need to keep track of childrenÊs progress. In other words, it is necessary to carry out appropriate from time to time. Preschool teachers are always concerned about effective assessments that can be carried out to assess childrenÊs understanding of mathematics. There are two suggestions. The first one is to prepare a proper documentation of childrenÊs work. Proper documentation of childrenÊs work provides teachers with the information about childrenÊs developmental progress. The second suggestion is to prepare a checklist for authentic assessment of each specific area to be assessed. The checklist should contain a list of sequential skills or learning outcomes that children are expected to achieve. Bhargava & Kirova (2002) provided three checklists for tracing and assessing childrenÊs development for three fundamental mathematical concepts: (a)

Matching and one-to-one correspondence (Table 3.6);

(b)

Sets and classification (Table 3.7); and

(c)

Order or seriation (Table 3.8). Checklist for Preschool Early Mathematical Concepts: Matching and One-to-one Correspondence

1. Matching 2 similar items 2. Matching two even sets of similar items with 5 items or more items for each set 3. Matching two uneven sets – with 5 or more items for each set

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4. Using appropriate vocabulary while matching sets (e.g. too many, not enough)

5. Matching two even sets of related items with 5 items or more items for each set 6. Matching two uneven sets of related items – with 5 or more items for each set 7. Using appropriate vocabulary while matching sets (e.g. too many, not enough) Adapted from Bhargava & Kirova (2002) Checklist for Preschool Early Mathematical Concepts: Sets and Classification

1. Sorting objects based on one physical attribute – colour, shape, size, material, pattern, texture

2. Classifying based on 2 physical attributes

3. Classifying based on 3 physical attributes

4. Describing or explaining what has been done when classifying by 1, 2, or 3 attributes

5. Classifying based on common functions

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6. Classifying based on association 7. Describing or explaining what has been done when classifying based on non-physical functions

8. Classifying by numbers Adapted from Bhargava and Kirova (2002) Checklist for Preschool Early Mathematical Concepts: Order or Seriation

1.

Comparing opposites (e.g., long/short, big/small, heavy/light, etc)

2.

Ordering three random order

3.

Ordering three objects by trial and error

4.

Performing reverse seriation

5. 6.

Performing double seriation Describing or explaining what has been done when carrying out tasks like comparing and ordering

objects

in

Adapted from Bhargava and Kirova (2002)

The activities designed by a teacher for the purpose of assessment need to match the knowledge and skills to be assessed. With appropriate activities, the use of the checklist will then be able to help determine as well as monitor childrenÊs progress and performance. It needs to be noted that assessments can be carried out in a formal manner as well as through observations of childrenÊs naturalistic and informal activities. It is to specify childÊs the dategrowth of assessment so as to be able to keep good record of useful each preschool and development in terms of amathematical concepts and skills. The individual checklist also helps to identify the individual kidÊs weaknesses so that assistance or remedial help can be provided.

TOPIC 3

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The concept of attributes forms the basis for the learning of many other higher mathematical concepts. The ability to carry out the task of classifying, matching, sorting and ordering is an important foundation for future learning of algebraic concepts. The learning of mathematical concepts and skills for children follows a sequential manner, which is in line with PiagetÊs stages of cognitive development. Informal learning is an important part of preschool childrenÊs education. ChildrenÊs progress and performance in mathematics can be monitored and accessed via proper documentation of childrenÊs work as well as using proper assessment checklists. Assessment can be carried out in both a formal and an informal manner.

Assessment checklists

Matching

Classifying

Ordering or seriating

Documentation

Sequential learning

Formal and informal assessment

Sorting

Kirova, A., & Bhargava, A. (2002). Learning to guide preschool childrenÊs mathematical understanding: A teacherÊs professional growth. Early Childhood Research & Practice, 4(1). Retrieved 10 September, 2013 from: http://ecrp.uiuc.edu/v4n1/kirova.html

To p ic

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4

Early Algebra Concepts: Patterns and Relationships

LEARNING OUTCOMES By the end of this topic, you should be able to: 1. Describe the relationship between patterns and algebra; 2. Identify and explain the different types of patterns that preschool children learn; 3. Plan activities for teaching and childrenÊs learning of patterns; and 4.

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Identify suitable assessments on learning of patterns.

INTRODUCTION

Algebra is a branch of mathematics that uses mathematical statements to describe relationships. When pupils in the secondary schools are taught algebra, they learn to use letters to represent numbers as well as to show relationships between numbers. For example, 5 + 2 is equal to 7 and 2 + 5 is also equal to 7, then the following mathematics statement can be formed: 5+2=2+5 Similarly, 3 + 4 = 4 + 3; 2 + 6 = 6 + 2; 1 + 2 = 2 + 1; and so on

TOPIC 4

EARLY ALGEBRA CONCEPTS: PATTERNS AND RELATIONSHIP

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If pupils are able to observe the above pattern, they will be able to understand that a + b = b + a where the letters a and b can be used to represent any two different numbers. Thus, it can be seen that the basic idea of algebra is all about patterns and relationships. Knowing that, it is then not difficult for us to understand why we need to teach young children about patterns and relationships if we want to expose them to early algebraic concepts.

4.1

THE IMPORTANCE OF PATTERNS

Basically, the first exposure of young children to the concept of algebra is pattern. Pattern plays an important role throughout the mathematics curriculum in childrenÊs education. NCTM (2000) highlights the importance of patterns in the following statement:

Patterns are ways for young children to recognise order and to recognise their world and are important in all aspects of mathematics at that level. Pre-schoolers recognise patterns in their environment and, through experiences in school, should become more skilled in noticing patterns in arrangements of objects, shapes and numbers, and in patterns to predict what comes next in an arrangement (NCTM 2000, p. 91) Patterns exist everywhere around us. Patterns exist in various forms. One example is the use of regular geometric shapes to form repeated patterns or tessellations for tiling purposes (refer to Figure 4.1)

Tessellations formed by repeated patterns

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TOPIC 4


Other examples are the use of numbers or attributes such as colours to form patterns (refer to Figure 4.2)

Patterns formed by numbers or colours

Before children are introduced to the concepts of patterns, they are exposed to activities like classifying, matching, sorting and ordering. These activities, together with patterning, form the basis for analytical and logical thinking, which are the essence of mathematical thinking. Engaging children in discovering patterns, recognising patterns, finding relationships between patterns, as well as forming patterns are learning and problem-solving experiences that are fun and interesting to children. Such learning experiences will certainly arouse the curiosity of children and engage them in the active learning process.

4.2

TYPES OF PATTERNS

Basically, there are three types of patterns that young children can learn. They are: (a)

Repeating patterns;

(b)

Growing patterns; and

(c)

Shrinking patterns.

4.2.1

Repeating Patterns

A repeating pattern is formed by continuous repetition of a core unit that is made up of a set of elements. In other words, it has a pattern unit that repeats over and over again. Figure 4.3, Figure 4.4 and Figure 4.5 illustrate the meaning of repeating patterns.

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A repeating pattern with a core unit of two elements

A repeating pattern with a core unit of three elements

Another repeating pattern with a core unit of three elements

In designing activities to help children understand the meaning of repeating patterns, the teacher needs to focus on helping the children to do the following: (a)

Identify the number of elements in each core unit;

(b)

Recognise the elements or objects in each core unit; and

(c)

Determine the relative position of each element in the core unit.

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4.2.2

TOPIC 4


Growing Patterns

In a growing pattern, the first unit is the core element. The subsequent units „grows‰ or „becomes larger‰ in a predictable manner. In other words, the pattern grows larger in the same way over and over again. Figure 4.6, Figure 4.7 and Figure 4.8 illustrate the meaning of growing patterns

A linear growing pattern

: A non-linear growing pattern

Another non-linear growing pattern

4.2.3

Shrinking Patterns

Shrinking patterns are essentially the opposite of growing patterns. In a shrinking pattern, the smaller‰ first unitinisa the core element. The „shrinks‰ or „becomes predictable manner. In subsequent other words,units the pattern grows smaller in the same way until it reaches the smallest possible unit. Figure 4.9 and Figure 4.10 are examples of shrinking patterns.

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61

A shrinking pattern

Another shrinking pattern

In designing activities to help children understand the meaning of growing patterns or shrinking patterns, the teacher needs to focus on helping the children to do the following: (a)

Observe the change between each unit and the subsequent unit; and

(b)

Compare each unit and its subsequent unit to investigate the relationship between them.

ACTIVITY 4.1 Prepare an activity sheet with exercises for children to extend each of the following patterns: (a)

Repeating pattern;

(b)

Growing pattern; and

(c)

Shrinking pattern

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4.3

TOPIC 4


CREATING PATTERN ACTIVITIES

As mentioned earlier, patterns can be found everywhere. But unless we are able to guide children to recognise patterns, it is difficult for them to explore and discover on their own the different kinds of patterns that exist around them. In guiding children to recognise patterns, we need first to make them understand that a set of objects or things that they have can be arranged in a repeated sequence. We need also to guide them to discover that there is not just one way, but a number of ways in arranging a set of objects of things or objects in repeated sequence. Through suitable classroom or outdoor activities, kids can be guided to create their own patterns. Patterns can be created with the help of children themselves or with the use of concrete objects. The teacher can also prepare and use manipulatives or pattern cards as teaching and learning aids. But there are many more ways to introduce children to patterns. Patterns can also be discovered through movements, sound, songs, music and art. Some of these will be discussed in the following sub-sections.

4.3.1

People Patterns

People patterns include patterns formed by arranging the people. For example, we can get children (Boys – B and Girls – G) to position themselves in a line following various patterns. Some examples of patterns that can be formed are illustrated in Figure 4.11.

Patterns can also be created using peoplesÊ movements or actions. Simple movement patterns such as „up, down, up, down . . .‰ or „left, right, front, back . . .‰ encourage children to use spatial orientations or physical movements as the repeating features.

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Some examples of actions and movements that form patterns are as illustrated in Figure 4.12.

4.3.2

Patterns with Concrete Objects and Pattern Cards

There is a wide range of categories of concrete objects that we come across every day that can be used to create patterns. Some examples of the categories of objects are: (a)

Cookies;

(b)

Fruits;

(c)

Boxes (of different sizes);

(d)

Blocks (of different colours);

(e)

Toys; and

(f)

2D geometric shapes.

Alternatively, we can create cards of various attributes (size, colour, shape, etc.) to form pattern cards. Figure 4.13 shows an example of the use of 2D geometric shapes to form a pattern. Figure 4.14 shows the use of numbered-cards to form patterns.

Using 2D-shapes to form patterns

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Using numbered cards to form patterns

In kindergartens, children are often taught to recognise patterns. Asking children to identify missing elements or objects in a pattern is a way to assess their ability to recognise the given pattern. Another way is to ask the child to extend a pattern by continuing to build the sequence in the same manner. However, to encourage creativity, children should be encouraged to create and form patterns of their own.

4.3.3

Creating Patterns Using Music

We can look at the role of music in the teaching and learning of patterns in two ways. Firstly, music and songs are built upon rhythms and thus follow certain regular patterns. As such, the learning of music and songs is an effective way to instil into children the idea of patterns. Secondly, many song writers for nursery rhymes and childrenÊs songs deliberately include some regular patterns in the lyrics to make them easier for children to learn. Let us look at the following childrenÊs songs and try to analyse how the songs can be used to help children develop the understanding of patterns (refer to Figure 4.15 and 4.16).

ChildrenÊs song: Old MacDonald had a farm YouTube - http://www.youtube.com/watch?v=7_mol6B9z00

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ChildrenÊs song: AAB Pattern YouTube - http://www.youtube.com/watch?v=o36gSj7AdZQ

ACTIVITY 4.2 Search the Web for two other songs that can be used to teach children about patterns. Then, design a learning activity that uses the two songs for learning about patterns.

4.3.4

Creating Patterns in Art

There are many connections between mathematics and art. In fact, mathematical principles have often been applied in the creation of art. Involving children in exploring and discovering patterns in art helps to promote creativity among children. Such learning activities also educate children towards appreciating the beauty of art.

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The following are examples of art patterns that children can form with the guide of the teacher:

Art pattern form using a single shape

Art pattern formed with a single shape but two colours

Art pattern formed using two geometric shapes, each with two colours

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Another art pattern formed using shapes of octagons and squares

It is interesting to note that the use of a single regular or a combination of regular shapes to form a pattern is the foundation of the creation of . are patterns formed when we cover a surface with flat shapes such that there are no overlaps or gaps. The creation of complicated tessellations requires a good knowledge and understanding of geometric principles and properties. Therefore, involving children in such art pattern activities represent a useful early exposure of geometry to children.

4.4

NUMBER PATTERNS

It is fine to expose preschool kids to number patterns. Initial exposure should however focus on the number pattern itself rather than teaching children how the patterns are formed and the relationship of the numbers in a number pattern. The initial concept of a number pattern is somewhat similar to their understanding of sequence in alphabets. For example, when preschool children are guided in counting 1 to 10, the focus is on the order of the numbers, that is, the number 2 is after 1, and the number 3 is after 2, and so on. It is similar to childrenÊs understanding that the letter B is after the letter A, and that the letter C is after the letter B, and so on. Children can only understand the relationship between numbers in a number pattern if they acquired the understanding of number sense and the basic operations such as basic additions and subtractions.

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4.5

TOPIC 4


CHILDREN AS CREATOR OF PATTERNS

According to PiagetÊs Theory of cognitive development, childrenÊs ability to learn is generally limited by their stages of cognitive development. The theory implies that when we provide learning for children, we should design it in such a way that they learn in stages, from easy to difficult. However, it may not be wise for us to rigidly put a threshold on what children should learn. In fact, the theory of constructivism emphasises that each child should be given sufficient opportunity to explore, investigate, discover, construct their own learning and grow at a pace that is suitable for him or her. After children have acquired some knowledge and skills about classifying, matching, sorting and ordering, they can actually be guided to apply such knowledge and skills in investigating and understanding the concept of patterns. Following that, children may then be guided to create patterns of their own. Before asking to create their own patterns, it may be good to ask children to extend patterns that we have started. In starting a pattern, we need to follow the following basic tips: (a)

A pattern can only be regarded as a repeating pattern if it is shown to be repeated twice or more; and

(b)

The exposure of patterns to children must be from simple to complex. (i)

For example, the following are two examples of AB AB patterns •

•

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(ii)


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And the following are examples of more complex patterns: •

ABC ABC pattern

•

AABB AABB pattern

Children need to be given the opportunity to „read‰ or „tell‰ others‰ the pattern that they have discovered or formed. They should also learn to „tell‰ others the relationship between patterns and relationship between objects within patterns that they have discovered. It is necessary to always encourage verbalising their thought through dialogue with peers as well as with the teacher. This, to the children, is an initial step towards concretising their mathematical thinking.

SELF-CHECK 4.1 Discuss the different activities you can use to encourage children to create the following patterns: (a) AABC AABC pattern; and (b)

4.6

XYYX XYYX pattern.

ASSESSMENT ON LEARNING OF PATTERNS

In assessing childrenÊs understanding of patterns, it is necessary to find out whether children are able to carry out the following tasks: (a)

Recognising patterns;

(b)

Describe patterns;

(c)

Copy patterns that others have made;

(d)

Extend patterns that other people have created;

(e)

Identify the missing elements in a pattern;

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(f)

Create their own patterns; and

(g)

Compare and talk with others about patterns.

Assessment of childrenÊs learning of patterns can be carried out in both the formative and the summative way.

4.6.1

Formative Assessment

Formative assessment can be carried out from time to time in an informal manner through observation while interacting with the children, or by assessing childrenÊs activity sheets. The teacher needs to be certain on what to observe and what to find out. In particular, the teacher needs to look for the following information: (a)

Are the children able to extend patterns?

(b)

Are they able to identify the similarities and differences between patterns?

(c)

Are they able to create patterns by drawing, colouring or even writing?

(d)

Are they able to use proper language to describe and explain patterns?

(e)

Are the materials to make patterns being used properly?

(f)

The teacherpurpose. should prepare suitable checklists or record sheets for assessment

4.6.2

Summative Assessment

Summative assessment is normally carried out upon completion of learning of a certain topic or a number of topics. For conducting summative assessment on childrenÊs learning of patterns, the teacher may do the following: (a)

Conduct both oral and written assessments to determine childrenÊs levels of achievements;

(b)

Prepare assessment assignments with exercises or activities ranging from simple to complex; and

(c)

Keep a record of the individual childÊs performance in each area in the form of an anecdotal report.

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ACTIVITY 4.3 Prepare an assessment checklist to assess childrenÊs understanding of repeating patterns.

•

The study of patterns and relationships is fundamental to learning of algebra.

•

Activities like sorting, classifying, ordering and patterning form the basis for analytical and logical thinking, which are instrumental for mathematical thinking.

•

There are three types of patterns that young children can learn: i) repeating patterns, ii) growing patterns and iii) shrinking patterns.

•

Children learn to form patterns using people, pattern cards, concrete objects, sound, songs, music and art.

•

Both summative and formative assessments can be carried out to assess childrenÊs learning.

Analytical and logical thinking

Formative and summative assessment

Anecdotal report

Growing pattern

Assessment checklists

Pattern

Create pattern

Repeating pattern

Extend pattern

Shrinking pattern

NCTM (2000). Principles and standards for school mathematics . Reston, Va.: National Council of Teachers of Mathematics.

Topic  Early Number Concepts

5

LEARNING OUTCOMES By the end of this course, you should be able to: 1. Explain what is meant by number sense; 2. Recognise activities that could be done daily by children that involve numbers; 3. Explain the concept of base ten and its application; 4. Describe the use of place value in multiplication and addition; and 5. Demonstrate how teach multiplication, subtraction and addition in 2-digit numbers asto well as in multi-digit numbers.

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INTRODUCTION

The foundation and basis of teaching mathematics to young children is knowing the concepts of number sense. Understanding number sense is critical in developing mathematical problem-solving skills. These involve counting skills, knowledge of numbers, estimation and the ability to use mathematical problemsolving strategies.

TOPIC 5

5.1

EARLY NUMBER CONCEPTS

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DEVELOPING NUMBER SENSE

Children are exposed to numbers and the use of numbers in their everyday activities. There are many daily activities that we may not realise involve numbers. These are some activities that are frequently done by children in their everyday life that involve numbers: (a) (b)

Arranging 4 candles on a birthday cake and blowing it one by one; Memorising daddyÊs and mummyÊs or friendsÊ phone numbers;

(c)

Changing TV channels using remote control to choose their favourite channel. For example, 103 to view TV3 or 611 to watch DisneyÊs;

(d)

Memorising mumÊs and dadÊs car registration numbers and also the teacherÊs car number when she came for home tuition;

(e)

While cooking with mum in the kitchen, mum asks to break 5 eggs in a big bowl, add 2 tablespoons of flour and a teaspoon of vanilla;

(f)

Knowing objects that exist in pairs such as eyes, ears, legs, hands, shoes, spectacles, pants and earrings;

(g)

Children are exposed to songs that involve numbers, such as ÊBaa Baa Black SheepÊ, ÂOne Two Buckle My ShoeÊ and ÂOver in the MeadowÊ; and

“One Two Buckle My Shoe” One, two, buckle my shoe. Three, four, shut the door. Five, six, pick up sticks. Seven, eight, shut the gate. Nine, ten, a big fat hen.

(h)

Children normally will start counting down the number of days to their holiday trip promised by their father or mother.

Number sense is the ability to recognise numbers, ways of representing numbers, relationship among numbers, and how to use those numbers so that every number has its value.

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5.1.1

TOPIC 5


Understanding Number Sense

Teachers should use the inquiry-based approach when teaching number sense to young children. As for children from a lower achievement group, explicit teaching of number sense is essential. The method of number sense guides the children to think wisely and not make drastic judgements which might be inappropriate. Having a developed number sense gives a person the ability to use and understand numbers such as: (a)

Knowing their relative values;

(b)

Knowing how to use them to make judgments;

(c)

Knowing how to use them in flexible ways when adding, subtracting, multiplying or dividing; and

(d)

Knowing how to develop useful strategies when counting, measuring or estimating.

Let us discuss an example. A 10 year old pupil is 1.5 metre tall. How tall will he be when he is 20 years old? Many children would answer Â3 metresÊ even though logically the tallest man on earth is found to be 2.51m tall (Guinness World Records, 2011). The reason children give such an answer is probably because: (a)

The children have less experience or none at all with the metre ruler; or

(b)

The children do not have the chance to learn while in the classroom and are not given the chance to ask questions.

Number sense is very much related to problem solving. Children that are encouraged to use them daily and given the opportunities to play with numbers will gradually develop mathematical thinking and an enthusiasm for mathematics. Number sense involves the knowledge on and . Let us look at two different representations of the quantity of eight. Figure 5.1 shows eight boxes placed side by side.

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: Eight boxes

One pupil may say, "There are eight boxes because I started counting from left to right and I count from 1 to 8 and stop.".

: Eight circles in two rows

Figure 5.2 shows 8 circles placed in 2 rows. A more advanced pupil would exclaim; „There are eight circles. The upper row has five circles and the bottom row has three. When I add 5 to 3, it gives me 8. If I have another 2 more circles than there will be 10 circles altogether. I also saw 6 circles in the first three columns and 2 more circles in the last two columns making them 8 circles altogether.‰ A ten-frame easily shows the arrangement and organisation of the basic-ten number system. The system provides pupils with the flexibility of individually counting each shape, or see the quantity of shapes as a combination of small groups of shapes. Children can also see a certain quantity related to ten. Most importantly, children are able to develop a range of strategies to calculate the quantity and understand the relationship between them. A pupil who is good in number sense usually: (a)

Excel in mental arithmetic and estimation;

(b)

Are skilled and flexible with numbers;

(c)

Are able to understand what the numbers mean;

(d)

Are able to see the world and make comparisons; and

(e)

Do not rely on algorithms.

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5.1.2


Number Sense Ability in Children

The development of children's cognitive mathematics has fascinated psychologists for years. Calculations and the relationships between numbers grow slowly in the early years. Children in preschool count manually without the use of any gadgets such as the calculator or any other means of counting instrument. This ability has three initial principles, namely: (a) Some children can count the number in correct sequence as 1, 2, 3, 4, 5, 6, 7 and so on. Later counting becomes unstable, 25, 40, 12. Yet a large number of preschool children are able to count from 1 to 10 (Baroody in Bideaud, Meljac & Fischer, 2013) (b) Children only need to specify one number, for example, when asked "What number?" Children will answer, "15" (c) Children know that they can count different objects such as various farm animals, cows, sheep, chickens, and ducks. These objects, however, need not to be the same and similar. The children also learn that they can count objects in sequence either from top to bottom, left to right or vice versa, in a circle or in a straight line. What is important is as long as no objects are added or reduced, the number remains the same.

ACTIVITY 5.1 1. Discuss with your friends how children use and practice numbers in their everyday lives. Do they memorise numbers or use objects in translating mathematical concepts of numbers? 2. Outline an activity that makes use of 2 or 3 pieces of board that had been divided into 10 equal squares on each board. Discuss and create an activity that will help children to be more proficient with numbers of either two digits or multi-digits.

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5.2


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MULTIPLE CONCEPTION OF PLACE VALUE

Pupils build understanding of place value by understanding the structure and sequence of the base ten system. In the base ten system, all the digits are organised in a particular way that is in multiples of ten. Understanding place value, however is more than knowing how to break numbers into hundreds, tens and ones but is linked to the knowledge of how a number is composed and its relationship to many other numbers.

5.2.1

Unique Features of Base Ten System

In preschool, the benchmark for place value refers to the use of numbers and operations in base ten beginning with 11 until 19. For beginners in preschool, the pupil needs to know that number 1 is not just a number 1 in the number 12, since 1 represents the place value 10 while 2 represents the place value of unit 1. This may seem to be a simple concept, but for a child to absorb the concept of place value in numbers, it might take more effort for the teachers to explain to them. Practically, Base 10 refers to the number system being used together. Let us take a number, 475 as an example. In the base 10 system, 5 refers to base one (unit), 7 is in place value ten (10s), while 4 is in place value hundred (100s). Every number is always ten times to the number next to the right of it. To simplify Base 10 let us refer to Figure 5.3.

Simplification of Base 10

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5.2.2

TOPIC 5


Place Value

Place value is used to determine a place of a number, whether the number should be at the hundreds, tens or at the unit place (ones). As the pupils proceed to a higher level, they will be learning more place values such as thousands, ten thousands and so on. The ability to understand place value is of great importance in assessing the mathematical skills in early childhood learning. Although there is a disagreement whether a child is capable of addressing problems in learning about place value, we insist to continue teaching them because of the following: (a)

Children are normally more attracted to big whole numbers such as 40, 60, 200, 900 and often ask about these numbers;

(b)

The numbers that are often heard and used by children do not reflect the basic structure of the place value;

(c)

Children who know counting up to the number 9 will not have difficulty understanding the value of these quantities; and

(d)

Quite a number of children, who are hesitant in mathematical calculations when reaching higher levels of education, lack hands-on experience in the system in a more concrete way.

5.2.3

Base 10 Block Concepts

Base 10 blocks are often used in early mathematics education to help pupils understand numbers. Base 10 has a cube to represent a strip of ten cubes to represent ten, and 100 blocks of cubes to represent 100. Base 10 also represents the power of the base 10. See Figure 5.4.

Base 10 Blocks http://math.about.com/od/glossaryofterms/g/Definition-Of-Base-10.htm

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Marker pens and a few pieces of paper with different numbers between 10 and 19.

(a) (b)

Ask pupils to make dots on the paper to represent numbers. (11 dots, 12 dots, 13 dots, 14 dots or 19 dots).

(c)

Then ask them to circle every 10 dots as shown in Figure 5.5.

(d)

Pupils will find that 16 is comprised of a group of 10 and 6, while 15 is comprised of a group of 10 and 5.

Group of ten : http://math.about.com/od/glossaryofterms/g/Definition-Of-Base-10.htm

(a) There are four unique features in the base 10 system, namely: (i) The system uses the values 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0. Zero (0) is a value that occupies a space. For example, 506 shows that there is no value in the tenth (10s) place or as representing a set of zero (null set/empty set). (ii)

Base 10 is a ranking system. The 'unitÊ is to the right. The next position is the base (b), and then the base (b) is multiplied again by the base (b). This sequence is repeated. (b = 10) ......b4 + b3 + b2 + b + 1

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432 = 4(102) + 3(10) + 2.

(iii) The total value of the number is the sum of its parts of the value assigned to each figure. (iv) We use this system to make all the basic computation using a procedure referred to as algorithms. This operation consists of the operations of addition, subtraction, multiplication and division. Base 10 decimal system is well used in the metric system for measuring. To assess the pupils' understanding, teachers may conduct classroom activities that can be made by the children themselves. Below is an example of an activity.

5 transparent plastic bag measuring approximately 15 cm × 10 cm. A bowl of any big sized beans (about 60 beans).

Dania has 5 plastic bags. Dania puts 10 beans in each plastic bag. After filling all plastic bags with 10 beans, Dania has 6 extra beans. How many beans does Dania have altogether?

Pupils must do the activity by using the materials provided to count the number of beans.

Children put 10 beans in all five plastic bags with 10 beans each. Children count that there are 50 beans from the plastic bag, which is 10 +10 +10 +10 +10. Children thethat 6 beans extra Children count find out Dania has outside a total ofthe 56plastic seeds. bag and add to the total of 50.

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ACTIVITY 5.2 Discuss strategies that could be used in the classroom to explain the concept of Base Ten and Place Value with your classmates in myVLE.

SELF-CHECK 5.1 Write the importance of introducing the base 10 system to the preschool children in a short essay.

5.2.4

2-Digit Number Concepts

Any whole number greater than 9 and less than 100 is a 2-digit number. For example, 11, 34 and 99 are all 2-digit numbers. The smallest is 10, and the largest is 99. 100 belongs to a 3-digit number. Fuson et al. (1997) have reported their findings and concluded that, there are five concepts in 2-digit numbers that children use. We will use the number 56 as an example to understand more about these 5 concepts. (a) Children count the numbers one by one: One, two, three, ...., ⁄, ⁄ ⁄, ⁄, five-four, five-five, five-six, which means 56. (b) Children are aware that it involves numbers in tens such as 30, 40, 50. However, after counting to the number 29, the child gets confused and begins mixing such numbers as 29, 41, 31, 60, 50. Children see 50 and 6 as 56. (c)

Children calculate in 10s, that is: 10, 20, 30, 40, 50, and later in ones, 51, 52, 53, 54, 55, 56. From here we could see that the understanding of place value is building up.

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(d) Children use the method of counting in units of tens, such as: (i)

one tens (1 x 10s);

(ii)

two tens (2 x 10s);

(iii) three tens (3 x 10s); (iv) four tens(4 x 10s); and (v)

five tens (5 x 10s).

And later start counting and adding a unit or sometimes even without mentioning the word twenty-something like, one, two, three, four, five, six. Fifty and six is 56 shows a good understanding of 2-digit value. (e) Children are fast in applying the concept of ten and unit. For example: Teacher:

Child: Teacher: Child:

Miss Maria has 56 pieces of biscuits. She puts the cookies in a box that can fit 10 cookies in each one box. How many boxes can be filled with 56 pieces of biscuits? Five. How do you count it? I know that 5 boxes will hold up to 50 pieces of biscuits.

The ability of children to solve problems depends on their ability to write and understand the concept of numbers when solving mathematical problems. There are two ways children could calculate the value in addition, subtraction, multiplication and division. They are through: (a)

The traditional approach; and

(b)

The childÊs own creativity.

Table 5.1 shows the mathematical problems involving two-digit numbers using the traditional approach and the creativity of a child.

TOPIC 5

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Addition, Subtraction and Multiplication in 2–digit numbers

Addition (+)

+

59 26 85

Subtraction (–)

‒

I add 9 and 6 making it 15. I place 5 at the bottom and bring 1 up to be added to 5 which gives me 6 then I add 2 to the number 6 and that gives me 8.

51 25 26

I will have to borrow 1 from 5 (10 since 1 comes from the ten place value), added to 1 now making the number 11 and then only I subtract 5 to get a balance of 6. At the place value tens I now have a 4 and I subtract 2 and that gives me 2, thus my answer

Multiplication (×)

×

33 45 165 132 1485

I will multiply 33 to 5 to get 165, then again I will multiply 33 to 4 to get 132. Lastly I will add 165 and 132 to get a total of 1485

is 26. +

50+20 = 9+6 =

59 26 85

–

70 15 + 85

51 25 26

25+25+1 = 51

×

33 45 1485

30+3 = 33 40+5 = 45

Then, I will take 25+25+1–25 = 26 25-25 = 0 25+1 = 26

I will start my multiplication as such: 30×40 = 3×5 = 40×3 = 30×5 =

1200 15 120 150 + 1485

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Table 5.2 shows the operation of addition using multi-digit numbers using the traditional approach and the creativity of a pupil. : Addition of Multi-digit Numbers

753 + 261 1014 I add 3 to 1 and obtain 4. 5 is then added to 6 to get 11 and I write 1 below and another 1 is carried to the top to be added to 7 which makes it 8. Then, I add 8 to 2 and that makes it 10. My final answer is 1014 753 + 261 753 = 700 + 50 + 3 261 = 200 + 60 + 1 700 + 200 = 900 50 + 60 = 3+1 =

110 4+ 1014

If children are given the freedom to choose their own working strategy and approach to solve mathematical problems, they will be more confident and will make a more accurate calculation when dealing with large numbers. Carrol and Porter (1998) felt that children should be given the opportunity to calculate the use of alternative approaches and create their own way. Children should be encouraged to count without over reliance on the calculator.

SELF-CHECK 5.2 Whatcan arethese the 5concepts conceptsbe that children could use in a 2-digit number and how used in the multi-digit number?

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85

Number sense is the fundamental concept of teaching mathematics to children. Having a sense and interest of numbers is critical in mathematical problem-solving skills. Skills in conducting basic operations in mathematics will improve the learning of mathematics. The ability to understand and use the number system is a very important asset in the development of numbers. Place value determines the value of a digit and the position of the number. The place value can be hundreds, tens and unit. As children proceed to a higher level in their education, they will encounter place values such as thousands, hundred thousands, millions and so on. Each one is 10 times greater than the next starting from the unit value. The concept of base-10 blocks teaches the pupils to make a collection in groups of ten. For example, 15 is composed of 10 and 5, 25 is made up of two tens and five (10 + 10 + 5). ChildrenÊs calculation will be faster and more efficient with a much more sound knowledge of the base-10 concept. Furthermore, the place value also uses the base-10 concept. The choice of selecting the appropriate approach to solving mathematical problems should be given to the children themselves. Understanding and frequent practice are better than memorising. Both approaches to addition, subtraction and multiplication for 2-digit numbers and multi-digit numbers, either traditional or by the creativity of the children themselves will empower the child.

2-digit number

One-to-one rule

Abstraction rule

Place value

Algorithm

Sequence tens and ones

Decade and ones

Stable order rule

Integrated sequence – separate tens and ones

Unitary – multi-digit Units of tens

Null set

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TOPIC 5


Bideaud. J., Meljac, C., & Fischer, J.P. (2013). Pathways to number: Children's developing numerical abilities. Guinness World Records (2011). Tallest Man – Living. Retrieved from: http://www.guinnessworldrecords.com/world-records/size/tallest-manliving

Topic  The Process of

6

Problem Solving

LEARNING OUTCOMES By the end of this course, you should be able to: 1. Describe the role of a teacher in teaching problem solving to young children; 2. Explain learning problem solving using the concepts of subtraction, addition, multiplication and division; 3. Demonstrate guiding young children to apply the easiest method of problem solving; and 4. Suggest ways in guiding young children to learn problem solving through multiple mathematical operations.



INTRODUCTION

Solving problems is part of the thinking process which is the foundation of young childrenÊs learning. Regarded as the most complex intellectual function, problem solving has been defined as a high-level cognitive process that requires the modulation and1987). control skills solving which isare moreofroutine or fundamental (Goldstein & Levin, Problem a form relief method involving certain processes that young children should learn in their early years of education. Thus, it must be valued, promoted, provided for and sustained in the early childhood classroom.

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TOPIK 6 THE PROCESS OF PROBLEM SOLVING

The process of problem solving is fundamental to the learning process of children in preschool mathematics. This foundation must be maintained in each classroom of early childhood learning. This process is found in almost all mathematical problems such as: seriation, classifying, shaping and arranging the sequence. This process is required because without it children may fall behind in learning to apply mathematics in their daily lives. They may fail to see the relationship between mathematics and their everyday life events. Teachers should focus on solving problems as part of understanding mathematics as children need to: (a)

Build new mathematical knowledge through problem solving;

(b)

Solve problems that arise in mathematics and in other contexts;

(c)

Apply and adapt a variety of appropriate strategies to solve problems; and

(d)

Understand the process of solving mathematical problems.

ACTIVITY 6.1 Discuss with your colleagues how learning mathematics can be applied in our daily lives, especially for children in preschool.

6.1

CLASSROOM EXPERIENCE WITH PROBLEM SOLVING

Many parents and teachers agree that the teaching of problem solving is a good start in the school curriculum. This curriculum fosters children to make decisions, to provide a framework for integrated learning, facilitates cooperative learning projects and organises work through themes and units. In instilling problem solving skills, teachers must trust childrenÊs attempts to learn through problem solving and maintain the classroom environment as one that encourages problem solving amongst the children. What are the strategies to employ in fostering problem solving in young children? What kind of classroom experience must teachers provide for these children to acquire problem solving skills? In this section, we will explore further about this.

TOPIC 6

6.1.1

THE PROCESS OF PROBLEM SOLVING

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89

Everyday Opportunities to Experience Problem Solving

Pre-schoolers learn best when they are given frequent opportunities to solve problems that are meaningful to them – what happens in their everyday lives. What teachers and parents need to do is to: (a)

Provide an opportunity for hands-on investigation. Offer children interesting materials to explore (like a magnet), finding objects and exploring malfunctioned (but safe) equipment can be interesting problem solving activities for the children. Change the materials frequently to ensure their minds are always engaged.

(b)

Foster critical and creative skills to encourage children to use the items in new and different ways. Strands of coloured beads, for example, can become the reins for horse racing, dollÊs hair, measuring tools or a tool to press into plasticine to make designs can also be used.

(c)

Encourage children to give suggestions and brainstorming by asking open-ended questions: "What can you do with ... ?" or "How many ways can you ... ?" Listen carefully to the ideas of the children.

(d)

Allow the child to find their own solutions. Offer assistance when they become frustrated, but do not solve the problem for them.

(e)

Use literature as a stepping stone in your method of teaching. Share a book showing how the characters solved a problem. For example, the story in the book titled King of the Playground by Phyllis Naylor and Caps for Sale by Esphyr Slobodkina. Children are easily fascinated by their heroes that they learn and read about through story books or stories narrated by the elderly people.

6.1.2

Guidance and Strategies to Foster Problem Solving Skills in Young Children

Teachers can foster young childrenÊs problem solving skills by providing enlarged blocks of learning time, rearranging classroom spaces to facilitate cooperation and providing a variety of materials. Nurturing a child to become a great problem solver is not difficult. There are four strategies that may be used as guidance for teachers to foster problem solving skills in children: (a)

Help children see mathematical problems around them,not only in the form of numbers, but also in the form of picture arrangements and cross-word

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puzzles, giving and following directions, and measuring and creating patterns. Game cards which involve a spinning wheel and dice are a good practice for children. (b)

Make sure there are a lot of modifiers, such as small blocks or coloured plastic materials that could be used as mathematical models in different situations. Fingers are also a good variable. In addition, children above 5 years should be given the opportunity to solve problems that arise daily.

(c)

Whether or not they use fingers, painting or mental strategies, ask the children how did they solve the problems. Use questions like, "How do you know?" "Can you show it to me?" and "Why?‰

(d)

Offer assistance only when necessary. A good strategy is to keep from helping too much as this would limit opportunities for children to solve problems. Use directions, instructions and gifts to encourage them.

If a child shows desperation, start by offering a little guidance such as "What do you think you can do?" It is surprising how a simple command like this can motivate children. If the guidance does not work, try giving instructions such as, "What did you do the last time we tried to measure how many scoops of cereals you eat for breakfast?" or "Do you think you can count the number on your other hand?" If necessary, communicate with the children by showing them strategies on how to solve the problems together.

ACTIVITY 6.2 Among the four problem solving strategies, which do you think is most effective and why?

6.2

ADDITION AND SUBTRACTION

We can teach children addition and subtraction by several kinds of methods, models and diagrams. The first step is to make sure the child knows how to count 1-10, 11-21 and able to tell the difference between numbers which are before and after, 10 is greater than 11 or 11 is smaller than 10. Next, knowing the sequence of numbers 5, 10, 15, 20, .... 100 is very useful when the children have started learning problem solving. The following are examples of mathematical problems for learning in early childhood education.

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91

Mathematics problems involving addition: (a)

Anna has 5 pieces of candy. She received an extra 26 pieces of candy from Hannan. How many candies does Anna have now?

(b)

Mariam has 2 nuts. Julia has 30 nuts. If Julia gives all her nuts to Mariam, how many nuts does Mariam have now?

(c)

Haris has 53 marbles. He is given another 16 more marbles by Shuib. How

(d)

many marbles does Haris have now? Mother bought 12 apples, 5 oranges, 6 mangoes and 10 pears for Faiz and his sisters. Daddy came home and brought 4 oranges and 10 apples. How many fruits are there altogether that Faiz and his sisters have?

Mathematics problems involving subtraction: (a)

Danial took 5 bananas out from a box containing 46 bananas. How many bananas are left now?

(b)

Nani has 75 apples. She gave 52 apples to her best friend, Nina. How many apples are left with Nani?

(c)

Rozi weighs 32kg, Amirul weighs 40kg while Nazmi weighs 42kg. What is the difference in weight between Rozi and Nazmi?

Examples of mathematics are order, in clear order. However, there are also more complexproblems problemsshown knownabove as mixed which includes the operations of addition and subtraction. Below are some examples of problems in the form of 'mixed order': (a)

There are 54 bottle caps in a box. Nurul has 8 bottle caps that she collected from school. She then put all her bottle caps in the box. Mawi came and took 10 bottle caps out of the box. How many bottle caps are left in the box now?

(b)

Mark has a huge box that is filled with 6 of his toy cars. He also has a small box that is filled with 3 of his toy cars. Mark decided to give away 4 of his toy cars to his brother Danny. How many cars are left in MarkÊs belongings?

(c)

Sheila has 10 strawberries. She gave 2 strawberries to her sister Susan. Sheila went back to her fatherÊs farm to collect 12 more strawberries. How many strawberries does Sheila have now?

Teachers usually choose the questions that challenge the minds of pupils. However, teachers must also not forget to wisely choose the questions that interest the children. When they read the question, children will feel excited to try and resolve the problem either by themselves or in groups. Avoid making

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questions that are stereotypes because children are easily bored if asked to do the same thing. There are four mathematical characteristics of equality in the operation of addition and subtraction (refer to Table 6.1): Knowing these characteristics can help teachers gather information about what is contemplated by the children while they attempt to provide answers to mathematical problems. : Mathematical Characteristic and Pupil's Rationale

(a) 0 is the identity element for addition:

If I have a million balls and I add 0 (zero) balls to my collection, I still have a million balls.

a + 0=a; 0 + a=a. (b) Any 2 factors a and b can be added and written in any order and will still give the same answer: ( a+b = b+a). (c) Whenever there are 3 or more whole numbers, any two of the numbers can be added first and after obtaining the total the third whole number can be added to it (a+b)+c=(b+c)+a.

(d) Any variable that has the same total number are similar to each other. For any set of whole numbers a,b,c,d,e,f, if a+b=c and d+e=f, therefore c=f. If c and f are the same whole numbers (5+2=4+3).

If I know that 7 + 4 =11, thus I also know that 4 + 7 = 11 because both follow the commutative property.

What is 6 + 8 + 2? Yes, by using the associative property I can first add 6+8=14, and I add 14 to 2 which gives me 16 or I can add 8 to 2 and get 10 thus I add 10 to 6 and get 16. *Children will use the associative property when there is a need to solve a series of values. They know where to begin and which number to add first and to add later. There are various ways in adding two digit numbers and getting a sum of 6. When I build a block design, I find that 4+2, 5+1, 6+0, 3+3 all sum up to 6.

: Reading Brick Mathematics for Early Childhood Education , OUM

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It should be noted that the identity element for subtraction is 0 (zero). There are two ways to prove the role of zero value in subtraction: 5 ‒0=5 and 5‒5=0. Children should be widely exposed to the identity element to avoid confusion during the later years of their math learning. For example, if I have 5 pieces of biscuits and I ate 0 pieces, I still have 5 pieces of biscuits. If I have 5 pieces of biscuits and I ate all 5 biscuits, I am left with 0 biscuits. Obviously 0 (zero) plays a dual role in this subtraction problem.

SELF-CHECK 6.1 What is meant by problem solving and how can a teacher teaching the pre-schoolers in early childhood learning nurture and encourage the pupils to excel in problem solving in mathematics?

6.3

MULTIPLICATION AND DIVISION

Multiplication and division operations in mathematics is interesting and quite challenging to the minds of the pre-schoolers until they proceed to a higher level. This process is linked to the problems experienced by pupils in their everyday lives. Thus this opportunity gives an option for children to be creative in their learning. Broad understanding of multiplication and division operations provides opportunities for teachers to encourage more problem solving in mathematics. The concepts of multiplication and division are not only interrelated but closely linked with addition and subtraction strategies as well as early fraction learning. According to Pepper & Hunting (1998) in Anderson and Thauberger, preschool children have the ability to understand the concept of dividing equally or dividing a collection into groups of same amounts. Children experience doing distribution or accumulation processes in their everyday lives. For example, one class has 33 pupils, the teacher asks the pupils to form the class into 3 equal groups. Pupils will begin to line up in three rows and each leader will count the number in each row. When there is a surplus in a row of pupils, the surplus pupil will be asked to move and occupy the row which is short of pupils. In mathematical learning amongst pre-schoolers, children will understand things much easier when it is done hands-on.

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Multiplication is the operation used to obtain results when two factors are known, example, (2 x 4) = 8. The operation is shown in diagrammatic form in Figure 6.1

: Multiplication (2x4=8)

Children benefit from the experience in classifying objects such as buttons or sea shells. For example, teacher may give this problem to pupils:

I have a bowl of blue buttons. In my bowl I have 6 blue buttons. If there are 3 bowls which contain as many buttons as mine, how many buttons do I have altogether? Interaction between teachers and pupils will make pupils become excited and learning becomes fun. Dividing operation is used when a child becomes skilled in multiplication. Division takes the total divided into several small groups. See Figure 6.2 for an illustration of a dividing operation.

: Divide operation (8 ÷ 2 = 4)

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In Figure 6.2, 8 buttons are being divided into 4 groups where each group contains 2 buttons. Figure 6.3 lists examples of mathematics problems involving multiplication and division operations.

: Maths problems involving multiplication and division operations

Each child thinks differently to solve a problem even when they are given a similar problem. Here is an example of some mathematics problems and the types of solutions:

Each bag contains 7 oranges. How many oranges are there in 6 bags? (a)

: The children use direct calculation by counting from 1 onwards until all items are counted.

The child counts 7 oranges. Next the child starts counting from the first set until all the 6 sets are completed with a total of 42 oranges. (b)

: The children use skip counting or skip counting in combination, to find a solution.

The child counts following the multiples of 7, such as, 7, 14, 21, 28,...while using the fingers. She/he stops at his/her 6th finger which is 42.

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(c)

: The child uses multiplication factor that he/she already knows to solve the problem.


I know that 7 5 is 35 therefore I just add another 7 oranges to give me a total of 42 oranges altogether, whereby 7 6=42. ×

×

(d)

: Children are able to use the facts given in a very efficient manner and are able to explain how the problem of a certain fact is interpreted. (Anghileri, Fennema, 1989; Carpenter, Ansell, Fennema, & Weisbeck, 1993; Carpenter, Franke, Levi,Franke, & Empson, 1999; Kouba, 1989)

"I know 7 6 is 42, so if there are 6 bags and each bag has 7 oranges, then the amount of oranges in all the 7 bags is 42", explained a child who uses a standard method of multiplication. ×

The thinking of a child is different for every child even of the same age. The level of intelligence of a child is subjective and difficult to measure, until they begin to enter a real school. At this stage of development the mind is evolving and growing. The latest technology has helped and played a key role in children's cognitive development. At an early age of 1 ½ years old children have already been exposed to iPads. The era of technology allows the impossible to be made possible and real. Table 6.2 describes the characteristics of mathematics and how it is being used by children. Mathematical properties include (i) identity of element (ii) commutative property (iii) associative property and (iv) distributive property : Mathematical Characteristics and PupilsÊ Rational

Number 1 is the identity element for multiplication and division. 1 a=a; a 1=a; a 1=a. ×

×

×

If I multiply or divide any whole number by 1, I will still get back the same whole number. (6 1=6; 1 6=6, or 6 1=6.) ×

×

When there are 2 factors, its arrangement during the process of multiplication is not important. The answer will remain the same (4 7=28; 7 4=28.) ×

×

×

If I know that 6 5=30, thus I also know that 5 6=30.This can be ×

×

provenobjects by making a sketch using or by seriation of objects.

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In multiplication operation involving 3 or more factors, two factors can be multiplied together before multiplying the answer to the next factor. As when a, b and c represent whole numbers, therefore: a (b c)=(a b) c.

If the volume of a box is 3cm 4 cm 5cm which is equal to 60cm3, I can multiply 3cm 4 cm=12cm2 and later multiply 12cm2 5cm=60cm3 Or I can muliply 5cm 4cm=20 cm2, then 20cm2 3cm=60 cm3.

In multiplication and division, we can treat the first factor as an addition to the next factor. As when a, b, and c represent whole numbers, therefore a (b+c)=(a b)+(a c).

If I can multiply 5 23 to give me 115, I can also multiply 5 20 =100, and 5 3=15. Later, I will add 15 to 100 and I will get 115.

×

×

×

×

×

×

×

×

×

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×

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×

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: Reading Brick, Mathematics in Early Childhood Education, OUM

SELF-CHECK 6.2 How is the process of multiplication and division linked to each other?

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Mathematical solutions which occur in many classrooms are meant to teach children to think critically and logically. Various strategies and methods are used in solving mathematical problems in preschool, depending on the suitability of the pupils themselves and their level of understanding. There are pupils who are more comfortable using fingers to count and there are those who use blocks and objects to solve mathematics problems. Pre-schoolers are introduced to simple addition and subtraction problems. These are called clear order problems. Sometimes they are introduced to more complex problems which are combinations of both subtraction and addition and are called mixed order problems.

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When the children are well versed in the problems involving multiplication, the children will be able to solve division problems. Knowledge in multiplication and division operations provide many advantages for teachers to teach mathematics to solve problems.

Addition

Division

Associate property

Identity element

Clear order

Mathematical standard facts

Cognitive process

Mixed order

Commutative property

Multiplication

Counting strategy

Subtraction

Direct model

Transitivity of equality

Distributive property

Variables

Early math: How children problem solve. Retrieved from: www.scholastic. com/.../early-math-how-children-problem. Goldstein F. C., & Levin H. S. (1987). Disorders of reasoning and problem-solving ability. In M. Meier, A. Benton, & L. Diller (Eds.), Neuropsychological rehabilitation. London: Taylor & Francis Group

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7

Organising Data in Mathematics

LEARNING OUTCOMES By the end of this course, you should be able to: 1. Explain data and data organisation; 2. Demonstrate guiding young children; and 3. Show assisting young children to apply the concept of probability in everyday life.

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INTRODUCTION

In the early stages of learning mathematics, pupils begin to learn about data and probability. This involves gathering information, organising information and analysing information. Pupils should begin learning to record data by using tally charts. There are two types of data: grouped data and ungrouped data. Children learn to record data and represent information in the form of tables, graphs and pie charts. Children will learn a variety of charts, graphs and tables. The graph used at this stage is a pictograph which uses pictures to present data values. Children will also be asked to interpret the data. Data does not only exist in the form of numbers and values. Data can also be collected in the form of an object that is being researched on. For example, a collected data about colours of balloons being released on TV3 Carnival: red, green, blue, white, black, yellow, purple, orange and gray.

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TOPIC 7

ORGANISING DATA IN MATHEMATICS

When probability is introduced to young children, they will be asked to compare the data using mathematical vocabulary, such as, „maybe‰, „maybe not‰, „possible‰ or „not possible‰. Questions on probability that could be asked include; "Is it going to rain this afternoon?" or "Can birds fly?" In this topic, we will see how you, as an early childhood teacher can introduce to children data and its organisation, the concept of probability and its application in daily life, as well as nurturing logical and mathematical thinking in young children.

7.1

USEFULNESS OF CHARTS

Charts and graphs are an important teaching tool in the management of data. Data can be transformed into charts to show patterns that could be easily discussed in classrooms. Children will begin to understand the process of analysing data when they interact with their surroundings. Children love to play games and they will spontaneously calculate and lay out their game, indirectly making mathematical calculations and data collection. Children should be given a variety of tools to synthesise, analyse and interpret information. Charts foster childrenÊs skills to make comparisons and obtain information. When calculating involving a huge collection of data, we need a way to interpret the data and the values obtained. The most effective and easy way to understand is through the use of charts and graphs. Among the advantages of the use of charts are: (a)

Simplifying calculation and rounding off numbers;

(b)

Decisions can be made quickly and accurately recorded;

(c)

Able to make conclusions based on the percentage;

(d)

Efficient and creative;

(e)

The quick way to see and evaluate objectives whether in the form of increasing numbers or decreasing values;

(f)

Clearly emphasise key points;

(g)

Convincing;

(h)

Compact way to present information; and

(i)

More interesting than just talking or printing.

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Nevertheless there are also some disadvantages in the learning method when charts and graphs are used, such as: (a)

It takes a long impoverished time because decisions must be made in advance to make a sketch, choose colours and materials;

(b)

It is more technical because pupils need knowledge to interpret and understand; and

(c)

It is expensive as it requires the use of tools such as manila cards, pieces of letters and others.

The srcinal purpose of the use of charts and graphs are: (a)

To demonstrate and compare changes;

(b)

To demonstrate and compare relationships; and

(c)

To state facts that become realities.

7.1.1

Varieties of Charts and Graphs

As discussed in the introduction, there are various ways to present data using charts and graphs. Especially for young children, attractive visuals interest them, thus the use of colours in charts and pictures in pictographs could entice them. The following are examples of charts and graphs used in the preschool mathematics learning based on an example of the data presented in Figure 7.1 regarding numbers of pupils and the total number of marbles collected.

Data for number of pupils with the amount of marbles collected

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(a)

Vertical Bar Graph

(b)

Horizontal Bar Graph

Vertical and horizontal column charts show that the longest column represents the number of pupils who collected the most marbles and the shortest column represents the number of pupils who collected the least number of marbles.

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(c)

: Pie Chart

A pie chart is also called a circle chart because of its round shape. A pie chart is often used to interpret data in terms of percentages. In this example, we mentioned that one student collected about 40 green marbles (32%), 20 red marbles (16%), 30 blue marbles (24%) and 35 purple marbles (28%). A pie chart collects and stores information in a way that impresses children. Although children do not see the numbers on the pie chart, they can see the breakdown of colours of the least and most abundant in the divisions of the circle. For example, in Figure 7.4, the child will interpret that the student collected the most blue marbles and least red marbles. (d) A pictograph helps children to analyse information easily from the chart. A pictograph is a form of a chart that uses pictorial objects to represent numbers. Figure 7.5 shows an example of a pictograph about varieties of apples in a food store.

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Pictograph

No doubt reading and interpreting charts and graphs would be a new challenge to the children. However, a graph plays a very significant role in the overall mathematics curriculum, providing a visual way of conveying information, for example, in functions or a set of data. Visual representations provides a picture frame and statistical information by combining symbols, points, lines, coordinate system, numbers, shading and colour with the purpose of conveying information fast and efficiently. Among the challenges faced by children in interpreting charts are their inability to interpret accurately due to their creativity in interpreting. Another difficulty is when pupils use software applications to create graphs. The software application is designed to enable pupils to visualise data to make arrangements of favourable information in space. However, a few non-interactive graphics software often produce graphs that are absurd and useless. Thus, as a teacher it is best to teach the children to use techniques that are suited to them rather than using the software package. Creativity in children builds their minds and thinking skills as well as helps them in problem solving.

ACTIVITY 7.1 Discuss with friends how graphs could be used in early childhood classes.

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ACTIVITY 7.2 A group of 100 pupils were selected to answer the question of whether they like outside of the classroom activities, do not like or are not sure. Data collected is as following: Likeoutdoor activities

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Do not like outdoor activities Not sure No answer

25 10 3

Present the following data into the form of vertical bar graph and pie chart.

7.2

EARLY EXPERIENCE WITH GRAPHING

Before graphs can be constructed, children must learn how to collect data. Data can be obtained by questionnaires given to respondents and their feedback provide us with data that we need. As shown in the following example, there are various forms of charts and graphs that could be drawn from the data. Example 1 shows the first step in making graphs beginning from data collection.

The following is an ungrouped data that shows the number of books brought to school by a child in Year 1 from Kelas 1 Permata in a class of 30.

4

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7

4

5 7

5

3

1

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5

7

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5

4

2

5

6 6

5 6 6

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:

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Gather the collected data and group the data collected using tally charts (see Table 7.1) to form a frequency table. Tally Chart

1 2 3 4 5 6 7

I I I

1 1 1

IIII I IIIIIIII IIII III IIII

6 9 8 4

Total number of pupils

30

Grouped Data

:

1

1

2

1

3

1

4

6

5

9

6

8

7

4

After acquiring the collected data, the children can begin to build a graph, chart or pictograph with the help of the teacher.

Figure 7.6 shows a vertical bar graph, showing the number of pupils who bring the various number of books to school, and constructed using Microsoft Excel. Whereas Figure 7.7 is the same data displayed through a pictograph.

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Bar Graph

A pictograph is a painting or a photograph in place of a value in a graph. In Figure 7.7, a picture of two books represents 2 books while a picture of a book represents a book.

An example of a pictograph

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7.2.1

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Mapping Techniques, from Pre-Graphic to Early Graphic

Young children aged 3-5 years old spend most of their time playing games and indirectly games involving activities such as collecting, counting, classification, and sometimes even the children in their own way make tally calculations. Teachers or parents can make use of this opportunity to get children to translate their activities to graphic presentations of numbers. Mapping techniques can be used to guide children in doing this. (a) A child can make the transition from the pre-graphic to an early graphic by using the strip boxes. Strip boxes are boxes representing a table that will be filled with graphics or pictures as counted by the child. Figure 7.8 shows an example of an empty strip box.

Rectangles drawn in 7 columns and 2 rows to make a strip box

(i)

Manila card;

(ii)

Sea-shells; and

(iii) Beans

(i)

On the manila card draw a rectangle with seven columns and two rows (see Figure 7. 8)

(ii)

A child is then given a pile of sea-shells and beans.

(iii) The child has to separate the sea-shells and place each of the sea shells in the drawn box on the manila card (see Figure 7.9).

: Six shells and seven beans arranged in rows

.

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(iv) Once all the shells and beans have been filled in the boxes, the children need to answer the questions that are asked by their teacher. Questions that might be asked by the teacher can be: (i)

What object is collected the most?

(ii)

What object is collected the least?

(iii) How many sea-shells are there? (iv) How many beans are there? or (v)

How many sea-shells are there more than the beans?

From the above example, the child will begin to learn ways of mapping and how to analyse and interpret data.

SELF-CHECK 7.1 Data collection is an important branch of mathematics. Discuss with your friends the benefits of collecting data during an investigation.

7.3

PROBABILITY

Probability is the likelihood that an event will occur or not occur. In statistics, probability of an event ranges from 0 to 1. 0 means that the event is certainly not going to happen, and 1 the event will definitely occur. Sometimes we measure probability in percentages. For example, "Chances that it will rain is 10% or you can use words such as impossible, unlikely, likely, chance, and sure. „It is likely to rain in the evening‰. In the early learning of mathematics for children, probability allows a child to think about the truth of an event. These are some examples of questions that involve probability and possible answers given by the children (see Table 7.3). Probability Questionnaire

Will tomorrowÊs weather be bright and sunny?

(a) Maybe (b) Sure (c) Not sure

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Is RM1.00 sufficient for a primary student to spend?

(a) Maybe (a) Might not

Can a 3 year old child drive a car?

(a) Impossible (a) Not sure

Cats like to eat durian.

(a) Not sure (a) Maybe (a) (a) (a) (a)

Tigers like to eat vegetables.

There are 101 green beans in the jar.

Impossible Impossible Maybe Not sure

(a) Not sure (a) Maybe (a) Yes

Questions involving probabilities can only be ascertained after the survey and calculation is conducted. Only then will a definite answer be known whether we have made a right or wrong assumption or hypothesis. The topic of probability, if taught using a proper technique is fun, and will be beneficial to pre-schoolers. Understanding probability will help children to think more critically and help them with other subjects. In Table 7.4 we are going to see some examples of using the concept of probability compatible to the early mathematical learning in pre-schoolers. Activities on Probability

A pupil throws a dice.

Perhaps number 1 will appear, or Perhaps number 2 will appear, or Perhaps number 3 will appear, or Perhaps number 4 will appear, or Perhaps number 5 will appear, or Perhaps number 6 will appear.

A pupils flicks a 50 cent coin.

Perhaps a tail will appear or perhaps a head will appear.

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3 books A, B and C are put on the table. By simply looking at the books, call the pupils one by one to guess (without touching the book) which book is the heaviest book and which is the lightest book.

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Maybe book A is the heaviest Maybe book B is the heaviest Maybe book C is the heaviest Maybe book A is the lightest Maybe book B is the lightest Maybe book C is the lightest

Sometimes, the most effective way to teach children is through games. Here is one example of a game involving probability. A black box is filled with several different marbles of different colours. Cut open at the top of the box a hole big enough for a child to put his hand inside to pick the marbles. Ask a child to put his hand inside the box, but before he takes out a marble, ask him what colour he thinks the marble that he is going to pick out from the box would be. Is it a white? blue? red? yellow? green? or an orange marble? A confident child might answer, ‰I am sure it is blue‰ or if feeling unsure the child might answer, „I think it may be a yellow marble‰. In probability, anything can happen. A 100% yes means the probability is Â1Ê and a totally wrong answer will have a probability of Â0Ê. Sample space is the different coloured marbles in the box. Another example that can explain the meaning of the sample space is: If in a kindergarten there are 500 pupils, but only 10 children are taken and asked to obtain their weights. Thus, Â10Ê is the sample space taken at random. The concept of probability can also be introduced to children in the form of matching with the correct information as shown in Table 7.5 below. Matching Probabilities

Maybe o Not impossible o Sure o

o o o

Elephants can fly Its going to rain tomorrow morning All children like to eat chicken

Not neccessary o Impossible o

o o o o

10 pupils will fail the final exam Aeroplane lands on the railroad Ships sail in sea and land The kitten likes to eat vegetables

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SELF-CHECK 7.2 Probability provides an opportunity for children to play with the questions and find the right answers after every investigation. Give some examples of questions that relate to the topic of probability in mathematics.

•

Children can learn to record data and represent information in the form of tables, graphs and pie charts.

•

Children who learn how to collect and organise data would be able to construct simple graphs such as bar graphs, pie charts or pictographs.

•

Probability in mathematics deals with a likelihood of an event. The probability ranges from 0 to 1. 0 (zero) means that the event is not likely to occur at all, and 1 means that the event is sure to happen.

Bar graph

Pie chart

Events

Probability

Grouped data

Sample space

Likelihood

Tally chart

Pictograph

Ungrouped data

Graf dan gambarajah – Scribdml. Retrieved from: http://scribd.com/doc/ 13265011/Graf-Dan-Gambarajah Mastura Badzis (2012). Reading brick mathematics in early childhood education. Open University Malaysia

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Mastura Badzis & Norzaini Yang. (2012). Study guide mathematics in early childhood education. Open University Malaysia. Math in Children's Literature, Data Gathering and Analyzing. Retrieved from: www.carolhurst.com/subjects/math/datagather.html Watson, J., & Fitzallen, N. (2010). The development of graph understanding in the mathematics curriculum.

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Measurement

8

LEARNING OUTCOMES By the end of this course, you should be able to: 1. Define measurement; 2. Differentiate standard and non-standard measurement; 3. Identify techniques in teaching measurement to young children; and 4. Demonstrate guiding young children to measure time.

 INTRODUCTION Measurement is the process of measuring the physical properties of an object by comparing it with a standard unit of measuring attributes such as length, width, weight or volume of an object or a quantity that is not physical in nature such as time, temperature, or the value of money Measurement can be done using non-standard means or standard measuring tools. Examples of measuring using non-standard means are using pencils, hands, legs or blocks to measure. These can be used as long as the tools used are not changed or altered. Examples of measuring using standard tools includes using a ruler, measuring tape and weighing balance. These tools use units of measure such as metre, centimetre, inch and others. Preschool is a good time to start learning simple measurement. To begin with, teachers could teach children about classifying things, for example, big, small, long, short, followed by using simple measuring tools. In this topic, we will explore how measurement is taught to young children.

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8.1

MEASUREMENT

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MEASUREMENT

In early childhood education, pupils use direct and indirect measurement and estimation skills to describe, compare, evaluate, plan and construct. The terms Ânon-standardÊ and ÂstandardÊ are often used when describing prescribed learning outcomes for measurement. Standard measurement uses ruler, measuring tape and other means of measuring devices with units such as centimetres or metres, feet or yards or inches. Non-standard measurement uses blocks, pencils, hands, feet, etc. as long as they remain unchanged. For example if we use an unsharpened pencil then the pencil should remain unsharpened throughout the process. Table 8.1 shows attributes in measurement in early childhood education. Attributes in Measurement

Comparative language for common attributes

• Use and respond to language associated with the attributes of length (long, short, tall), capacity (full, empty), mass (heavy, light) and time (before, after, oÊclock, days of the week) • Use and respond to the comparative language of: – – – –

Choose units to match the attribute to be ÂmeasuredÊ

•

length (ÂMy dog is shortÊ) capacity (ÂI have more water than you because my cup is biggerÊ) mass (A brick is heavier than an egg) time (ÂWe have lunch before we go to playÊ)

Some objects/events are better to use as units than others (for length choose a unit that has ÂlengthÊ like pencils, for capacity choose a unit that is easy to use such as a cup) things are measured for a purpose (measuring the ingredients for a recipe)

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MEASUREMENT

Non-standard units are used to measure length, mass and capacity

Directly compare length, mass, capacity, time and area

Time is measured

• Non-standard units can be used to measure: – how long (use hand spans, paperclips, popsticks, straws) – how heavy (use blocks, marbles, counters, washers) – how much something holds (use cups, cans, jugs) • Use the same units when comparing objects or events (use popsticks when comparing the length of a desk with the length of a bookcase) • Use the same units when comparing objects or events (to find out how heavy each rock is, use the mass of a marble as the unit and count the marbles needed to balance each rock and compare) • Strategies to directly compare objects and events by different attributes: – length, width, height by lining up one end – mass by lifting (holding one object in each hand to determine which is heavier) – capacity by pouring – time by starting at the same time (find who can pack up the blocks the fastest) – area by placing one region over the other • Identify and sort objects according to comparative language (put all the small animal counters together; find which rock is heavier; arrange three paper fish from smallest to largest) • The same object can be ordered differently according to different attributes (a beach ball may be larger than a basketball but a basketball is heavier) • The same objects can be ordered differently according to different attributes (a long glass is taller than a shorter glass, but may not necessarily hold more water than the shorter glass) • Particular things happen at particular times (ÂAfter lunch we have a rest; before bed we read a storyÊ)

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MEASUREMENT

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• Clocks are used to tell the time and the hands move or digits change as time passes. Read key times (oÊclock, half past, 9 oÊclock – school starts, 12 oÊclock is lunch time and 3 oÊclock is home time) on analogue and digital clocks Everyday language of approximation when judging size

• Everyday language associated with approximation (nearly, about, just over) make size judgements in everyday situations: – ÂI think I am tall enough to reach the light switchÊ (length) – ask ÂWho do you think could balance you on the see-saw?Ê (mass) – choose a piece of paper from the front of the room that is big enough to cover a desk (area) – choose a box that all of the balls will fit into (volume) • Estimate length by looking (this pencil looks longer than that pencil), compare by lining up the ends • Find things that are clearly bigger or smaller than the object (ÂI am taller than the bookcase and shorter than the doorÊ)

: Department of Education and Training Western Australia, Early Childhood

ACTIVITY 8.1 Discuss with your colleagues on how to devise an activity on measurement for young children.

8.2

PRINCIPLES OF MEASUREMENT

Measuring an object requires knowledge of the procedures and standard units and the properties of estimates. Measurement will explain the concepts of numbers which are not rational, the characteristics of circles, formulas for area and volume.

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MEASUREMENT

There are several measurement principles that should be known when measuring, such as: (a)

When taking measurement of an object, make sure the measurements are done repeatedly to ensure accuracy for each measurement. This would eliminate errors due to the measuring instrument or error done by the person himself;

(b)

Measurements usually start from the starting point (0). If measurements begin in the middle of a tape measure or ruler, we must reject the values which are not used;

(c)

Formal and standard measurements use different units, metres are used to measure the length of a high jump pole, the centimetre unit is used to measure the length of a pencil. A two-dimensional shape, example a square or a rectangle, uses unit square (cm 2 or m2) for its measurement of area. Volume of a 3-dimensional cone is measured and the unit used is cubic unit (cm3 or m3);

(d)

Two pieces of strings may be tied up together and its length is then measured. Example, the first string measures 20cm while the second string measures 15cm, thus after tying the two strings together the new length recorded is 32cm. However we should not be adding two different objects with different volumes. For example, a cup of sugar and two cups of water does not make 3 cups of sugar or 3 cups of water;

(e)

During each measurement there will be some slight error, but to obtain zero error a more precise unit and measuring device is used. Example, recording the running time of a runner. We record the time in units of minutes, seconds and hundredths of seconds. Runner A might have a time of 2 minutes, the same as runner B, but runner A shows a time of 2 min. 15 seconds. Without the „second‰ measurement of time, runner A seemed to be running with the same time as runner B; and

(f)

Measurement is transitive. For example, if a book (object A) is 11cm long, a pencil (object B) is 11cm and the length of a piece of paper (object C) is also 11cm. Since all objects have the same size (11cm), so we can assume that the measurement of A = B, A = C and B = C. Through measurement, it can tell us that objects A, B and C have the same size without the need to organise the objects next to each other to find out whether the objects are of the same length.

Undoubtedly, children are able to acquire knowledge measurement as early as in preschool until they enter standard 1 and begin to learn the techniques of measurement in a more complex way.

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In fact, children have a high level of imagination. Parents and teachers should not prevent children from exploring their imagination but must also help develop their imagination towards betterment. Some examples of preschool children's imagination: (a)

A child pretendingto be a doctor: "Your forehead is hot, you're sick. I will give you some medicine."

(b)

A child pretendingto be a shopper in a supermarket: "I want to buy a kilo of beef. How much is a kilo of beef?"

(c)

A child pretendingto be a mother: "This milk is hot, wait until itÊs cold then only you can drink it ".

(d)

A child pretendingto be a chef: "Put 500gm wheat flour and a quart of milk."

At this age, childrenÊs imagination should be allowed to grow and not be constrained, as in time children will learn to recognise and relate to the concept of measurement. Spontaneous games are more meaningful than those taught and instructed by teachers.

ACTIVITY 8.2 Discuss in detail the principles of measurement and try to explain what error may occur during activities of measuring.

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FORMAL APPROACHES AND TECHNIQUES OF MEASUREMENT

Young children might find it difficult to count and calculate and are more prone to learning the concept as a game. Thus, as an introduction to measurement, children could use simple techniques with wood blocks or fingers. Figure 8.1 shows how blocks are used to measure the length of a scissors. In the figure, the scissors measures 9-blocks. The same block is also used to measure the width of a book.

Measuring using blocks

The use of standard measuring tools and real materials train the children to practice the correct way to measure other objects, making assumptions on the size of an object seen or used. We use measurements not only to measure the length or width of the object, but also to measure and calculate area and volume of objects, time and temperature. Table 8.2 shows classification of the types of tools to measure different objects and the units used.

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Classification of Tools and Unit Measurement of Objects

Pencil

length

ruler

cm

Paper

area

ruler

cm2

Box

volume

ruler/measuring tape

cm3

Car Room

length area

measuring tape measuring tape

m2 m

Liquid (water)

volume

measuring cylinder

l/ml3

Book

weight

weighing machine

g/kg

Children

weight

weighing machine

kg

Ice/hot water

temperature

thermometer

C

cm= centimetre cm2= centimetre square cm3=centimetre cube m=metre l=litre ml=mililitre g=gram kg=kilogram C= degree Celsius Measurements are divided into two categories: (a)

A quantity which consists of length, width, height, area, volume, capacity, weight, and mass; and

(b)

A qualitative attribute which consists of the measurement of time, temperature, and money.

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Figure 8.2 are examples of tools of quantitative measurements.

Examples of measuring instruments

Despite the excitement of a child to learn the techniques of measuring and calculating the values of the measurements, there are some difficulties encountered by children. This could be due to several reasons such as: (a) Children are easily deceived by looks. An object is said to be heavy just because it is big. A large ping-pong ball is said to be heavier than a small rubber ball;

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(b)

The next difficulty that may arise is the concept of units, and how to use the unit. Children do not realise that using a smaller unit will increase the number of units in measurement of a fixed quantity. For example, a piece of string measuring 1 metre length is 100cm, 2 metres is 200cm (centimetres). Nevertheless 1m=100cm and 2m=200cm;

(c)

The error can also occur when the child does not place the object at point '0Â on the ruler;

(d)

Complexity also occurs because children are not able to distinguish measurements of area and perimeter;

(e)

Although measurement is a technique of measuring objects to discover its inequality between objects, children are more likely not to use the correct measuring devices , but more to just making comparisons. For example, ‰I have a bigger cake than you‰, „I'm taller than you‰, „My pencil is longer than your pencil‰; and

(f)

The illustrations in textbooks are rather vague and confusing for pupils in the early childhood learning especially in the use of units such as cm and inches.

SELF-CHECK 8.1 1. Formal approach of measurement uses instruments suchwatch. as a thermometer, weighing scale, measuring tapes and stop How can the children be taught the techniques of measurement in a formal way. 2. What are the constraints that might occur when using a formal approach in your teaching of measurement to young children?

8.4

VOLUME, WEIGHT, LENGTH AND TEMPERATURE

Volume, weight, length and temperature are quantitative measurement. In early learning of mathematics, children use informal measurement with only the fingers, paper clips, blocks or pencils. For example, the length of an exercise book is 15 paper clips. Even before rulers could be introduced to children, teachers must ensure they are familiar with numbers from 1 to at least 50. Rulers are used to measure length, width and height. Once these measurements are obtained, they can calculate areas, perimeters and volumes.

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(a)

Measuring area and perimeter of a rectangle

Figure 8.3 shows a rectangle measuring 5cm long and 3cm wide. The area of this figure is obtained by multiplying its length and its width. (5cm 3cm) which gives a yield of 15cm2.

×

The perimeter of Figure 8.3 can be obtained by adding all the measurements around the rectangle. In a rectangle, the length of AB is equal in length to the length of the CD, as well as the length AC = BD. Therefore , the perimeter is the total when adding all sides (5cm +5cm +3cm +3cm) which is equal to 16cm. (b)

We can calculate volume for 3-dimension objects such as cubes or cuboids.

Measuring volume of a cuboid

The solid cuboid in Figure 8.4 has a length of 5cm, 2cm width and 3cm height. The volume of this block can be found by multiplying the values of length, width and height to produce the volume. [(5x2x3) cm = 30cm3]. Therefore the volume of the cuboid is 30cm3. Pre-schoolers may be confused with the concept of calculation on volume. To facilitate learning of volume amongst children, small cubes can be inserted into the block and then count the number of cubes.

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When told that a bottle contains 1 litre of mineral water, young children will only give a blank stare. Why? Because 1 litre does not carry any meaning to a child. A more practical and easy way for children to understand is by getting the children to make a comparison. Compare a bottle of mineral water 1000ml/1 litre, with 500ml milk (half litre) and a bottle of 'VITAGEN' 25ml. Set up three plastic containers of the same size and shape. Pour the mineral water in the first container, milk in the the second and finally pour 'VITAGEN' into the third container. Children will notice that the amount of mineral water (1 litre) is more than milk (500ml) while VITAGEN '' (25ml) is very little compared to the rest (refer to Figure 8.5).

Different Volumes of Liquid

(c) Young children often hear adults say 'heavy'. "Basket is heavy, do not lift", "HaniÊs school bag is heavy, let mummy carry it to your class", "How heavy is this papaya?". Teachers can create their own scales using hangers and paper cups (refer to Figure 8.6). Children will be excited to use these scales.

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: A simple weighing device or scale : http://www.howdonkey.com/pics/52/make-a-balance-scale-for-kids.jpg

Teachers should explain to the young child that the cup contains the object tilted down because it is much heavier than the object in a second paper cup. Encourage children to use a variety of objects and record the weight of the two objects that are weighed. Through this activity the children will realise that a large object does not necessarily weigh more than the smaller object and small objects might be heavier than bigger objects. If the child is able to read and recognise letters and numbers, the actual scales can be used. There are various tools to consider, depending on the type of the object to be weighed (refer to Figure 8.7).

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Types of weighing machines

(d) Teachers are encouraged to make a chart for weather and temperature, and paste it on the walls or notice boards in the classroom (refer to Figure 8.8).

Example of a weather chart http://res.cloudinary.com/demo/image/fetch/w_120,h_200,c_scale/ http://www.adaycare.com/Images/WeatherChart2.jpg

In the early stages, the young children will be taught whether the weather is hot, cloudy, rainy or windy. Later, they are taught that the temperature

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during hot weather is higher than the temperature when the weather is cloudy or raining. As the children begin to understand the differences in weather and temperature and the different environmental conditions, the teacher may introduce tools that measure temperature, that is, the thermometer. There are 3 types of thermometers: (a) (b)

Clinical thermometer to measure the temperature of the human body; Laboratory thermometer to measure the temperature of the liquid in the laboratory; and

(c)

Room thermometer to measure the room temperature and the surrounding environment.

Different types of thermometer

To make learning more interesting for the children in the classroom, teachers could bring a clinical thermometer to the children. A simple activity can be conducted. The teacher randomly chooses some pupils and take the temperature of their bodies. A special thermometer placed on the childÊs forehead, is used. An example of the childrenÊs temperatures is recorded in the data in Table 8.3

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Data on childrenÊs body temperatures.

Amirah

37.0

Faizal

37.3

Amin

37.0

Aaron

37.2

Abigail

38.0

Our normal body temperature is 37ÀC. From the data obtained, we may notify the pupils that AbigailÊs body temperature is 38.0ÀC, a little higher than the normal body temperature. Thus, there is a possibility that Abigail is having a slight fever. Every day on television, weather reports display data for the states in Malaysia and overseas. Young children should be encouraged to watch it, and parents should try to explain the concept of temperature to their children.

8.5

INFORMAL MEASUREMENT: THE CONCEPT OF TIME

Young children can feel how heavy a bag of peanuts weighing 500g and a box of raisins weighing 100g are. They can easily tell which object is heavier and which is lighter. Children can know which pencil is longer and which is shorter but they find difficulty in understanding the concept of time and need adults to assist them on the time concept. Learning through songs about the days of the week is one of the ways to learn about time. Children love to sing! Learning becomes fun and easy through music and poetry. Children learn faster when the days are noted on the calendar while singing songs or reading a poem. Thus, encourage children to have their own calendar. A variety of information can be recorded on the calendar each day. Parents would be surprised to see many activities, and plans made by their children. In the concept of physics, time is categorised as a qualitative and informal measurement. Time involves a period, or how long and a sequence. Two aspects of measurement that will be discussed are time and day. Children are taught to tell time and specify the period of time.

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The curriculum of early childhood education begins to teach the children to look at pictures of clocks and tell the time shown in hours: 1 oÊclock, 2 o'clock, 3 o'clock and 12 o'clock, besides describing the concept of night and day which uses the same count. Once the child is able to read the time on the clock according to the count of the hour, children will be taught how to count and read the movement of the minute which is calculated every 5 minutes. Now, in modern technology, children are exposed to a digital clock where they only need to read the numbers displayed on the screen to tell time. For example, 3.47am or 4.19pm. The activities in many classrooms provide opportunities for pupils to learn the concept of time effectively. Some examples of activities are shown below:

By using a stopwatch, the teacher tells the pupils to close their eyes for 1 minute. After the one minute, stop the stopwatch and asked the pupils to open their eyes. This activity is designed so that pupils can feel the duration time of 1 minute. The same activity can be repeated for 2 minutes or 3 minutes.

Children are given a drawing of an object to be coloured or painted. Before starting to colour, the children were asked to look at the clock and note the time . Once the children have finished colouring, they will be asked to note down the time. The children will learn and notice how long they took to finish colouring.

All children are given a checklist for them to make a note of the time taken while making a number of activities, daily for a week, as set out in Table 8.4. Checklist on time taken for activities

Journey to school from home. Doing school homework at home. Watching ÂDoraemonÊ on television. Helping mother wash dishes after dinner. Going to the zoo with family members.

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With such a schedule, the child will always feel excited to create activities and constantly think about time.

SELF-CHECK 8.2 Children sometimes have difficulty in doing the process of measuring. What are the techniques that could be used when measuring liquids using informal measurement?



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There are two types of measurement for young children; standard and nonstandard measurement. Standard measurement uses specific measuring instruments and devices following National Standard Unit. Non-standard measurement uses either fingers, elbows, blocks, pencils or even paper clips to measure objects and uses the technique of comparing the differences in lengths or weights of two or more different objects. Teaching measurement using proper techniques avoids significant error when measuring area, perimeter or when measuring the volume, weight, temperature and time.



The concept of time is very important to preschoolers because it involves our everyday lives, no matter how young or old. Time affects our lives and children can be trained to appreciate time and use it well.

Centimetre

Stop watch

Error

Temperature

Measuring cylinder

Thermometer

Measuring tape

Volume

Standard measurement

Topic  Mathematics

9

Learning Environment for Early Childhood

LEARNING OUTCOMES By the end of this course, you should be able to: 1. Identify the structure of social and emotional learning of pupils in learning mathematics; 2. List various instructional strategies to employ in mathematics learning amongst young children; and 3. Apply suitable and effective methods in mathematics learning for early childhood education.



INTRODUCTION

Developmental stages in children gradually develop and expand as they grow older and maturity in thinking increases. Learning mathematics is not conducted only in formal schools but should begin in preschool. This is because the effective learning of a child develops at an early age, and will form a strong foundation of learning that will help them to continue

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learning at a higher level. Based on lessons planned by teachers, children as young as pre-schoolers can become fascinated in the learning of mathematics. Therefore, teachers should play a role that will attract children to gain interest in mathematics, and not let that interest fade away as they grow older. A strong basic knowledge in mathematics is important, as our society is moving towards a developed nation founded on Science and Technology. Our pupils should have a strong understanding of basic mathematical concepts that will enable them to pursue their studies at a higher level. To ensure the future of our nation, we need to review and adopt a more effective method and strategy in teaching children the basics of mathematics in their early schooling.

9.1

MANAGING THE LEARNING ENVIRONMENT

The classroom is one of the most conducive places for learning activities. A lot of time is spent by teachers and pupils in the classroom. The environment in the classrooms that is created and encouraged by the teacher determines whether the pupil's ability to learn is increasing or decreasing, and whether it is conducive for the children to learn and play. Some things that teachers can consider in managing the learning environment include: (a)

Building an environment that is able to provide effective teaching and learning;

(b)

Strengthening the role of teachers in the mathematics learning environment;

(c)

Developing the integration of technology in teaching and the learning environment; and

(d)

Organising teaching strategies to improve the learning environment.

Apart from having a comfortable and conducive learning centre, the materials in teaching mathematics in early childhood learning must also be carefully designed to diversify teaching, in order to attract the attention of children.

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Other than that, the learning environment is also affected by factors such as: (a)

The belief of teachers teaching mathematics;

(b)

Teaching expertise in mathematics and an understanding of how pupils learn mathematics;

(c)

Past experiences of pupils;

(d)

Expectations of parents; and

(e)

School policies.

The activities in mathematics learning also plays an important role. Activities must be appropriate with the learning environment. Activities that may be conducted include the following: (a) Activities in the classroom should involve all pupils doing the same activity. Normally the activities are based on information presented by teachers through instruction and narration, while question and answer sessions will be done at the end of the sessions. (b) Group activities are very important because through this activity children learn to the share and take participate. When conducting a group activity, teacher mustturns maketosure that every boy and girl is involved. Groups should be in small numbers so that all children are able to engage or participate. Teachers can choose same or different activities for each group. The main objective is to develop the child to work together and cooperate with one another in solving the problem given. (c) In addition to the activities in the classroom and in groups, the teacher should also provide individual activities. There are times when individual activities are suitable for children because they provide the opportunity for children to be independent, and to allow teachers to give instruction to the individual child. Each activity must be planned so as to avoid wasting time and can thus provide the maximum benefit to the pupils. The teacher should plan learning activities in the classroom which should last for half an hour (30 minutes). Plan activities which involve the number concept. Table 9.1 shows an example of activities suitable for 5 year old pupils at the kindergarten.

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Example of a Mathematics Lesson Plan for 5-year old Pupils

Mathematics 20 pupils 5 Year old 9am until 9.30am. (1/2 hour)

9.00am-9.05am Introducing cube, cuboid and concept of numbers. Teacher explains while showing the block. 9.05am-9.15am

Provide sense of numbers and experience to the children on the concepts of number and quantity. Children are exposed to using blocks of various shapes. Let the children hold two blocks, one on the left hand and one on the right hand, saying "a block". Every time they show the block, they will be repeating the words but the number will be increased, „two blocks‰, three blocks‰, four blocks‰, and so on. Over time, the child can arrange the blocks in accordance with the characteristics of the other.

9.15am-9.25am

To develop the ability to count in sequence and in a position such as first, second, third and so on. Children take a few pieces of the block. Each person count the blocks one by one. Other children will be watching. 1. Several pieces of blocks arranged in a row and the children will be counting the blocks. 2. The blocks are compiled into a tower. Children count the blocks used to make the construction. 3. The blocks are arranged in a straight line. Children are asked to show the position of the first block, second, third, and so on. These activities are conducted after the child is familiar with the concept of numbers. In the initial stages do not exceed the number 5 then 10.

9.25am-9.30am

Giving a sense of fun to learn mathematics. Teachers sing with children a song that has numbers in the lyrics while using fingers. Over time the child can count with the fingers while singing.

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ACTIVITY 9.1 1. Plan activities in the classroom suitable for children aged five years. Describe in detail how you plan to execute the plan in your own classroom. Write your answer in myVLE forum. 2. What preparations should you do as a teacher teaching mathematics to preschool children?

9.2

SOCIAL AND EMOTIONAL STRUCTURES OF LEARNING

Social and emotional learning provide an opportunity to advance education towards a positive approach, rather than a punitive one to promote healthy adolescent behaviour. Moreover, learning is also intended to develop a more robust learning ecosystem to appreciate emotional and social intelligence. An excerpt from „School-Based Social and Emotional Learning Programmes‰ identifies some elements of emotional learning skills. The elements are listed in Table 9.2. Socio-Emotional Learning Skills

Self-Awareness

• Recognising and knowing one's emotions • Understand the causes and circumstances of a person • Recognising and knowing the emotions of others • Recognising the inner strength, and stir positive feelings about self, school, family, and support network • Knowing the needs and values of a person • Looking at the requirements and values of a person • Believing personal effectiveness • Have a sense of spirituality • Value diversity

Social Awareness

• Showing respect for others • Listen carefully and accurately • Increase empathy and sensitivity to the feelings of others • Understanding the views and feelings of others

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Responsible for decisions

• Analyse the situation perceptively and identify the problem clearly • Practice social decision making and problem solving skills • Respond constructively in solving interpersonal problems • Engage in self-assessment and reflection • Face up to personal responsibility, morality, and ethics.

Relationship Management

• Managing emotions in relationships, harmonising the various feelings • Demonstrate sensitivity to social-emotional cues • Expressing feelings effectively • Communicate clearly • Involve others in social situations • Build relationships • Collaborate • Practicing self-tendencies, leadership, and persuasion • Manage conflict, negotiation, and refusal • Preparing and looking for help

Self-Management and Organisation

• Expressing and dealing with anxiety, anger, and depression • Controlling impulses, aggression, and self-destruction and antisocial behaviour • Manage personal and interpersonal pressures • Focus on existing tasks • Setting short and long term goals • Plan carefully and thoughtfully • Modifying performance based on feedback • Mobilising positive motivation • Activating hope and confidence • Work towards optimal performance

„School-Based Social and Emotional Learning Programmes‰

SELF-CHECK 9.1

1. Give examples of how negative emotions and stress can affect learning of a pupil. 2. What are the elements of socio-emotional learning skills and the effects on the learning of the pupil?

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TOPIC 9


INSTRUCTIONAL STRATEGIES FOR ENHANCING MATHEMATICS LEARNING ENVIRONMENT

The learning in any subject will change from time to time in accordance with the development of technology. Now, use of computers and high-tech materials have dominated the market and become one of the teaching materials and mode of learning. As such, various methods have been developed by experts in education so that the level of education in Malaysia is on par with other developed countries in the world. The following are some instructional methods and strategies that can be employed by teachers to enhance mathematics learning environment: (a) Induction is used to obtain conclusions, facts or general characteristics of research on some specific mathematical model. In this method, pupils should study mathematical examples, make comparisons and analysis before arriving at conclusions. Figure 9.1 shows the learning process through method of induction which could be practised to enhance the learning of mathematics.

Learning process through induction method

(b) Deduction method encourages the use of formulas, laws or mathematical theorems that have been studied, to arrive at a conclusion or new mathematical generalisation. With this method, pupils can gain vast experience and knowledge of mathematics enough to arrive at the formula, law, or new mathematical theorem. Figure 9.2 shows a deduction method used in obtaining theorems from formulated mathematical laws.

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Learning process through deduction method

(c)

This method includes the planning of all activities, investigating, analysing and recording findings. The objectives of this method are to: (i)

Help pupils know the things they want to know;

(ii)

Expand pupils' knowledge; and

(iii) Promote critical and creative thinking among pupils. Figure 9.3 shows steps used in the inquiry-discovery method starting from identifying the problem until a summary or solution is achieved.

Inquiry-discovery method

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(d) Discussions encourage children to communicate with one another and to learn to work well in groups. This enhances their collaborative skills and also nurtures them to be team players. These are the objectives of using the discussion method: (i)

To challenge oneÊs thinking;

(ii)

To stimulate attitudes and beliefs;

(iii)

To build skills of interacting with others;

(iv)

To promote mental development; and

(v)

To train pupils to think more thoroughly.

At the beginning, the teacher will start a discussion and ask questions related to a topic. During the discussion, pupils are encouraged to give their views while the teacher acts as an observer, stimulating pupils to give opinions and help pupils make the conclusion. Focus areas in discussions that teacher must take into consideration are: (i)

The relevance of the discussion;

(ii)

Accuracy of facts;

(iii) Relevance and fitness of ideas presented; and (iv) Mode of discussion. (e) Questioning children can encourage critical thinking in them. The objectives of the questioning method are to: (i)

Assist pupils to recall and reinforce concepts or skills they have learned;

(ii)

Guiding pupils to participate actively in learning;

(iii) Guiding pupils to use their brain power to solve problems; (iv) Generate interest and curiosity in pupils; and (v)

Helping pupils revise their mathematics.

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This method is powerful in probing childrenÊs understanding of what have been learnt. However, the teacher must use certain questioning techniques to gain quality responses, and for positive learning experience. Below are some guidelines that may help mathematics teachers in mastering techniques of questioning efficiently and effectively: (i)

Using mathematical language appropriate to the level of achievement of pupils;

(ii)

Build simple, accurate, clear and contained questions only for a concept or a fact;

(iii) Use questions that encourage pupils to think; and (iv) Use the right questions according to teaching objectives. Figure 9.4 shows four characteristics in the questioning method that can be used by teachers.

Characteristics of questioning method

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Table 9.3 elaborates the characteristics of the questioning method. Characteristics of the Questioning Method

Frequency

Equal distribution

Initiator

Waiting time

• Refers to how often and frequent the questions posed by the teacher during the learning and teaching process • The frequency of questioning encourage more pupils to participate and improve their study achievement • Teachers should not ignore the weak pupils and focus only on the smart ones • Teachers should avoid questioning all pupils at one time • Teachers are not encouraged to call the pupil's name before asking, for fear that only pupils whose names are called will be thinking • Teachers ask questions or give commands, to get information or answers from pupils who have failed to respond or respond incompletely • This is a quiet scene while waiting for a reply • Waiting time for an answer to be given should be quite a short; less than one second • The benefits of waiting time are: – The lesson can run smoothly and is focused – Quality and quantity of pupilsÊ responses also increases – Failures to answer decline because pupils who voluntarily participate to answer increase

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Figure 9.5 shows the stages of questioning according to BloomÊs Taxonomy.

Stages of questioning according to BloomÊs Taxonomy

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Table 9.4 shows the examples of questions that could be used in the process of questioning based on the levels of BloomÊs Taxonomy. Examples of Questions According to the Levels of BloomÊs Taxonomy

Knowledge

Understanding

Memorise Fact Terminology Definition Concept Principle Understanding of the meaning of material

Usage

Choosing a concept/skill and use to solve problems

Analysis

Dividing

the

material

into

sections and relevance

explain

its

Assessment

Making a decision based on the criteria before the proof

Synthesis

Produce something pure after dividing materials into specific components

What ....? List .... Name .... Define Describe ..... Explain ... Simplify ... Give an example ... Translate ... Predict .... Solve .... Use .... Construct ... Calculate .... How can these data be used? Why ...? How? ... Linked with ...? What is the difference ... Between ... and ... Criteria for ...? What results can you do ...? Compare and contrast ... Criteria for ...? How do these data support ...? What can be predicted as a result of data

(f) Practical work is normally done by the children who are given tools to carry out their own experimentation while the teachers act as facilitators, facilitating and guiding the children while the children undertake the practical work. Through this method, children can discover the principles, mathematical theorems with resources or tools of mathematics. For example, pupils may

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be asked to build mathematical models that will give them a concrete mathematical concept that is abstract. (g) Expository is a way of delivering education through information, storytelling or demonstration with the aim of teaching. Teachers provide information in advance and pupils listen carefully to understand and remember them. Figure 9.6 shows the stages of delivering information to the pupils in the expository method.

Expository method

Table 9.5 summarises the advantages and disadvantages of using the expository method in teaching. : Advantages and disadvantages of expository method

• Teachers save time, easy to control discipline • Effective in conveying information, concepts and principles of mathematics

• Boring • Becomes a long lecture if used in the whole of teaching and learning concept

(h) When children study mathematics, they need concrete activities to understand the symbols and signs of abstract mathematics. In the laboratory method, pupils use mathematical resources or tools available in the laboratory to experience concrete activities while discovering the principles, laws and theorems of mathematics. For example, to find out the relationship between the measurement of units such as 1kg = 1000g, 1L = 1000ml, 1m = 100cm, children may get hands on experience by using measuring instruments such as scales during laboratory sessions.

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(i) It is a technique that involves teachers demonstrating how to do an activity or steps in front of the pupils. Demonstration starts with teachers demonstrating what needs to be done while pupils hear, observe and attempt to do the activity. For example, a teacher can demonstrate how to draw an angle using a protractor. Some of the advantages of using the demonstration method are: (i)

Pupils will be able to emulate techniques or skills accurately and easier, when trying on their own;

(ii)

A good demonstration strengthens pupilsÊ memories on the steps to be taken;

(iii) It facilitates pupilsÊ understanding and creates interest in learning; (iv) Pupils will undergo practical training to build a strong and precise skills; (v)

The teacher is able to correct any mistake as it happens and thus, pupilsÊ performance can be enhanced;

(vi) It enables pupils to be more confident to perform an individual demonstration; (vii) Ideas, principles or concepts can be illustrated during demonstration; (viii) It reduces time consumption compared with a trial and error method; and (ix)

Pupils will not perform unguided activities and thus avoid accidents.

(j) Cooperative and collaborative learning is a teaching strategy in which pupils interact in a small group, with a purpose and a common goal. Pupils work together to learn and are responsible for the learning of their team mates, in addition to their own learning. Collaborative learning is based on cooperative learning. To create a collaborative classroom, one easy way is to conduct cooperative learning where the teacher can control more variables in the class. When pupils become more familiar with how to work together, depending on each other to acquire knowledge, then they will grow to be collaborative learners.

TOPIC 9


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Similarities between cooperative and collaborative learning are: (i)

Emphasises active learning;

(ii)

Teachers act as facilitators;

(iii) Teaching and learning experienced by pupils and teachers; and (iv) Enhances the cognitive skills; Pupils are required to take responsibility for their own learning: (i) Allows pupils to present their ideas in small groups; (ii)

Develops social skills and team building;

(iii) Helps increase pupilsÊ success and knowledge memory; and (iv) Utilises pupilsÊ diversity. Table 9.6 shows the differences between cooperative and collaborative methods. Differences between Cooperative and Collaborative Methods

• Pupils receive training in teamwork and social skills • Activities are structured where each pupil plays a specific role • Teachers monitor, listen and intervene in group activities if necessary • Pupils need to submit their work for assessment at the end of the lesson • Pupils evaluate the performance of individuals and groups under the guidance of teachers

• It is assumed that the pupils have the social skills required. They need to develop their existing skills to achieve learning goals • Pupils make discussion and self-organise • The group activity is not monitored by teachers. If any questions arise, questions are answered by the group itself. Teachers only guide pupils toward resolving the question • Pupils keep working draft for further tasks • Pupils evaluate the performance of individuals and groups without the guidance of a teacher

(k) In this method, teachers are the main source of instruction and information. Pupils passively receive information with the goal of acquiring information passed by the teacher. The teacher controls the situation and act as the leader. The focus of teaching or activity is monopolised by the teacher.

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TOPIC 9


Some examples of teaching strategies that are used in teacher-centred approach are: (i)

Story telling;

(ii)

Lectures; and

(iii) Demonstration. Figure 9.7 shows the characteristics of teacher-centred approach in teaching and learning.

Characteristics of teacher-centered approach

(l) This approach focusses on: (i)

Active learning and increase pupilsÊ involvement;

(ii)

Pupils have the freedom to engage in activities with teachers;

(iii) Teachers act as facilitators and are responsible for the pupil; and (iv) Learning emphasises in independent learning, self-discovery and inquiry-discovery method.

TOPIC 9


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Figure 9.8 shows the characteristics of pupil-centred approach as a way to increase the pupilÊs involvement in learning.

Characteristics of pupil-centred approach

(m) In media-centred approach, teachers use many media or materials such as books, building blocks, painting materials, scientific equipment, computers, resource, self-access centres and others in carrying out the teaching. Use of materials in the learning and teaching techniques should be based on several factors such as learning objectives, pupilsÊ needs, instructional strategies and abilities of pupils. Various types of media or materials used during learning and teaching are able to improve the learning and teaching techniques and can enhance critical thinking and mental skills. Some of the types of teaching and learning media are: (i)

Slides and filmstrips;

(ii)

Transparency projectors;

(iii) Tape recordings and television; (iv) Three-dimensional models such as globes and human skeletons; (v)

Organised modules; and

(vi) Learning packages using computer.

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TOPIC 9


SELF-CHECK 9.2 What are the instructional strategies used to enhance the learning environment in mathematics?

•

Learning mathematics in the early stage of schooling needs to be planned carefully as at this stage, the child will begin to recognise numbers and looked forward to learning mathematics.

•

The activities must be arranged according to the age of the child, and have a variety of activities and presentations that are interesting for further learning.

•

Social and structured emotional learning aims to develop a more robust learning ecosystem to appreciate the emotional and social intelligence.

•

A number of strategies are used to enhance the learning of mathematics among pupils, such as, induction, deduction, inquiry-discovery, questioning, expository, laboratory and cooperative and collaborative methods.

Cooperative and collaborative

Induction

Cuboid

Laboratory

Deduction

Self-awareness

Demonstration

Self-management

Expository

Social and emotional learning

School-Based Social and Emotional Learning. Retrieved from: http://www.rci. rutgers.edu/~melias/Handbook%20of%20 Child%20Psych.pdf

Topic  Assessment of

10

Mathematics in Early Childhood

LEARNING OUTCOMES By the end of this course, you should be able to: 1. Explain the purpose of assessment; 2. Explain the foundation of assessment in relation to pupilsÊ age group; 3. Describe the role of assessment in early childhood education; 4. Select suitable assessment strategies to assist teaching in order to enhance the quality of childrenÊs learning; and 5. Define evaluation analysis results.



INTRODUCTION

In the normal schools, assessments and evaluations are done every semester to assess the childrenÊs competency. In preschool, evaluations and assessments are also conducted on the early childhood learners to determine their achievements and progress, so as to diagnose any problems in the child or any problems that might arise in the method of teaching and learning. It will also help in making decisions on matters related to the curriculum. In doing so parents will be notified of their childrenÊs progress

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 TOPIC 10 MATHEMATICS ASSESSMENT IN EARLY CHILDHOOD

Mathematical foundations found in preschool include communication, social skills, and other positive skills from the aspects of physical, cognitive, emotional, spiritual and aesthetic values. Mathematics in preschool is not just a study where the children are asked to memorise numbers, but understanding the value or meaning of the numbers. During the assessment of mathematics, several contents are identified such as concept of numbers, identifying numbers and figures, and plus and minus operations. Children are tested on their understanding of the problems in mathematics and ways of solving the problems. Assessment for early childhood must be developed and practiced in the best interests of children and their future in the learning of mathematics in particular. Evaluation is important because it can assess the cognitive development of the intellectual, emotional, language and communication and behaviour of a child.

10.1

ASSESSMENT OF MATHEMATICS IN EARLY CHILDHOOHD

Children of similar ages often know different things. For example, some young children in kindergarten do not recognise any numeral from 0 to10 but, about one third of these children already recognise all these numerals. Some even know many two-digit numerals. Most children beginning Grade 1 count by ones accurately, and know a few skip counting sequences, but some children are still coordinating one-to-one counting of objects. The way to determine childrenÊs learning needs is to assess what each child already understands in mathematics, and to find areas where each child begins to experience difficulties. Table 10.1 is an extract from Early Childhood Assessment in Mathematics Manual. The development of mathematics in a child occurs gradually as they grow from age 4 to 9, and each child will develop a different skill from another child of the same age group. A child who has typically mastered counting small numbers will respond to some addition problems such as 2+2, 3+3 or a 4+4. Assessments areofdone based on strands. the foundation of learning of numerical different stages mathematical The strands used in this interview and are separate. This is because children often have different combinations of understandings in mathematics. One child may be very strong in Numeration, but has only an early understanding of Geometry. Another child of similar age

TOPIC 10 MATHEMATICS ASSESSMENT IN EARLY CHILDHOOD

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may have a strong understanding of Geometry, but is less advanced in Numeration. Foundations for Assessments

(a) Beginning

These children are just beginning to learn the „counting by ones‰ sequence. They can recite small sections of the counting sequence. When counting objects, they are just beginning to coordinate one object with each number name.

(b) Early counting

These children are confident counting to ten. They can recite the counting sequence beyond ten, and can coordinate counting objects with number names to at least ten. They can use materials to show a given number less than ten.

(c) Counts objects

These children are proficient at counting objects by ones. They know the counting sequence to at least thirty. They can coordinate counting objects with numbers names beyond twenty.

(d) Counts one by ones

These children can mentally manipulate the „counting by ones‰ atsequence. They than can one, start and the count counting sequence numbers other one by ones. They can count backward by ones from ten.

(e) Skip counts

These children are beginning to count in various intervals (skip count). They can skip count forward by twos, fives and tens. They can count backward by ones from twenty.

(f) Facility with counting

These children are proficient with counting in various intervals. They can skip count by various numbers, and can count by fives and tens starting at any number. They can count backward by ones from larger numbers.

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(a) Beginning

These children are just beginning to recognise numerals. They may name some numerals from zero to ten. They do not yet understand the order of these numbers.

(b) Understanding to 10

These children know all numerals from zero to ten, and understand the correct order of these numbers.

(c) Exploring to 100

These children are beginning to recognise and compare many two digit numerals. They may confuse similar „sounding‰ numbers such as thirty and thirteen. They can tell the number that comes after a two-digit number.

(d) Understanding to 100

These children are beginning to understand the place value of a two-digit number. They understand that ten ones are grouped to form one ten. They can order two digit numbers, and can tell the number before a given two-digit number.

(e) Understanding to 999

These children are able to recognise three-digit numerals. They have a growing understanding of the place value of three-digit numbers. They can compare and order numbers to 999.

(f) Understanding 1000 and beyond

These children can recognise and order numerals beyond 1000. They show a solid understanding of place value.


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(a) Beginning

These children are just beginning to understand the concepts of putting small collections together, or taking a small amount from a collection. They are not yet successful in finding sums or differences.

(b) Count all strategies

These children are just beginning to understand the concepts of putting small collections together, or taking a small amount from a collection. They are not yet successful in finding sums or differences.

(c) Count on/count back strategies

These children are beginning to use „short-cuts‰ to add two small collections, or to take a small amount from a collection. They only need part of a model, and either „count on‰ to find a sum, or „count back,‰ „count down to,‰ or „count up from‰ to find a difference.

(d) Initial addition/ subtraction strategies

These children are beginning to use „short-cuts‰ to add two small collections, or to take a small amount from a collection. They only need part of a model, and either „count on‰ to find a sum, or „count back,‰ „count down to,‰ or „count up from‰ to find a difference.

(a) Beginning

These children are just beginning to understand how to make small equal groups, and how to share a small amount equally. They are not yet successful in finding products or dividends.

(b) Full model/count all

These children can make small equal groups, and can share small amounts equally. They can find products by counting all objects by ones from one. They see a group of objects only as a collection of ones.

(c) Full model/count sets

These children can make small equal groups, share small amounts equally, and divide a collection into groups of a given number. They use skip counting or repeated addition to find products. They see a group as „one unit‰ as well as a collection of ones.

(d) Part model/count sets

These children need only part of a model to find products or dividends. They use skip counting or repeated addition to find products and dividends.

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(e) Multiplication strategies

These children are able to multiply two numbers, using thinking strategies such as multiplying by tens, doubling, repeated addition, or „turning numbers aroundÊ. They are able to mentally multiply small numbers without making a model.

(f) Division strategies

These children are able to divide a number equally, using thinking strategies such as dividing by tens, halving, and using repeated addition or subtraction. They are able to mentally divide small numbers without making a model.

Beginning

These children are just beginning to match very simple two-dimensional figures shown in conventional orientations.

Whole shapes

These children can name and match simple twodimensional figures shown in conventional orientations. They match shapes to common „prototypes,‰ e.g. isosceles or equilateral triangles only. They attend to the whole of the shape, and do not yet describe any part of a shape.

Exploring shape parts

These children are beginning to attend to the parts of a shape, and typically describe the numbers of sides and corners a figure has, after counting these. They can visualise rotations of some simple shapes.

Understand shape parts

These children attend to many parts of a shape, and are able to fully describe a figureÊs properties. They recognise examples beyond common „prototypes,‰ e.g. scalene and right triangles. They can visualise rotations and reflections of some simple shapes.

Shape families

These children understand how shapes belong to families, e.g. a square is a special type of rectangle, and also a special type of rhombus. They attend to parallel lines, angle and perpendicular lines. They are able to visualise rotations and reflections of many shapes.


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Beginning

These children are just becoming aware of the attribute of length. They can find the longest object perceptually.

Compares objects

These children can compare the length of two or more objects. They know to alignthe objects to compare their lengths. Theyhow can identify longest and the shortest object.

Uses informal units

These children can use informal units to measure and describe the length of an object. They understand how to repeatedly use a constant sized unit without gaps or overlaps.

Uses formal units

These children can use one system of formal units to measure an objectÊs length (either Standard or Metric, but not both). They are able to reasonably estimate a length.

Facility with formal units

These children can use both the Standard and the Metric systems to measure an objectÊs length. They understand fractions of these units.

Beginning

These children are just beginning to read numerals on clock faces.They do not yet link these numerals to units of time.

Hours/ days/month

These children can tell time to the hour. They know the days of the week and the months of the year.

Half hours/dates

These children can tell time to the half hour, and can find dates on a simple calendar.

Tells time

These children can tell time to five-minute intervals, and can use a calendar to predict dates.

Facility with time

These children can calculate using hours and minutes. Early Childhood Assessment in Mathematics Manual

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ACTIVITY 10.1 Discuss in a group how an evaluation system is able to improve the teaching of children in the early childhood environment.

10.2

ROLE OF ASSESSMENT

Assessment is important because it encompasses all the decisions that we make on our children when we teach them. All the overall results of the evaluation will lead and transform the learning outcomes of children. The assessment should be part of the learning process and is carried out continuously for the benefit of the children as well as the educators, where information on the development and progress of children are gathered using a variety of methods. This assessment is intended to identify the various intelligence and potential to be strengthened and enhanced. The assessment of the progress of every preschool child in the cognitive domain, affective and psychomotor should be accurately and clearly recorded continuously. Studies from several previous research defined evaluation as ‰the systematic process of determining educational objectives, and collecting, using, and analysing information on the learning, to make decisions about programmes, individual pupil progress, or accountability. Therefore, evaluation is not just a test at the end of every instruction given by the teacher but a guide to the teacher in decision-making patterns and techniques. Most of all, assessment should bring benefit to the child. There are 6 roles of assessment that will allow the child to be evaluated fairly and in a most efficient way, including: Assessment should benefit the children. Gathering accurate information from children is difficult and potentially stressful. Assessment must have a clear benefit – either indirectly to children or in improving the quality of educational programmes. Assessment should be tailored to a specific purpose and should be reliable, valid, and just for that purpose. The assessment which is designed for one purpose is not necessarily valid if used for other purposes. In the past, a lot of test abuses against children have occurred because of misuse of assessment techniques. Policy evaluation should be designed upon realising that the reliability and validity of assessment increases with the age of the child. It is quite difficult to


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make an accurate assessment of a child who is of a very young age until the child reaches the age of 6. Thus, some kind of assessment should be postponed until the child is more mature, while other types of evaluation can be performed, but with more caution. Assessment should follow the appropriate ages in the content and methods of data collection. Assessment of young children should include early learning and development network, that fully includes physical well-being and motor development, social and emotional development, approach towards learning, and development of language and cognition, and general knowledge. The method of assessment should recognise that children need to be able to exhibit normal context of their abilities. Tasks that require the use of paper-and-pencil can be difficult for children to demonstrate their abilities and understanding. The assessment should use appropriate language. This is because regardless of whether the assessment is designed to measure early reading skills, knowledge of colour names, or learning potential, the results might easily fail due to language proficiency. This is especially for children who have limited exposure to the English language if the evaluation is done in English or otherwise. Early language development and second language of every child should be taken into account, when determining the appropriate valuation methods and in interpreting the results. Parents should be an important source of assessment information, as well as people around them. Direct actions can cause errors in the evaluation, therefore, the assessment should include multiple sources of evidence, particularly reports from parents and teachers. Results of the assessment should be shared with parents as part of an ongoing process that involves parents in their child's education. Evaluation of teaching and learning is as important as teaching and learning itself. The main goal of the assessment is to improve teaching and learning. Thus, valuation or assessment should be conducted to enable teachers to: (a)

Detect the development of children as a whole and assess the extent of pupils' understanding of the subject;

(b)

Look at the changes in the development of each child;

(c)

Make the planning and renovation of instructional activities to help enhance the growth and development of children;

(d)

Study the needs of each child;

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(e)

Detect the effectiveness of activities, routines, and strategies used;

(f)

Identify the strengths and weaknesses of pupils in their learning from time to time;

(g)

Provide accurate reports to parents, health experts, teachers and monitoring officials;

(h)

Appreciate the work of children;

(i) (j)

Make immediate follow up; Evaluate the effectiveness of instructional techniques that were presented by the teacher;

(k)

Make basic decisions about what is and is not suitable for children; and

(l)

To determine the effectiveness of programmes and services that the child received is beneficial and appropriate.

The teachers must be sensitive to the various technologies that are available for evaluation purposes in a variety of situations. These technologies can help teachers in developing the test, making the screening test and review the results. Evaluation is important because it can assess the cognitive development of the intellectual, emotional, language and communication and behaviour of a child. This assessment includesdiagnostic, assessing standard, alternative evaluation, detection evaluation, detection assessment development, performance assessment and psychological assessment. This assessment can be done to a child from birth to the end of their life. Evaluation can be done step by step. The types of evaluation can be categorised as shown in Table 10.2 Types of Evaluations

Standard Assessment

Standard tests used to compare data across class, school, district and state levels. Many standardised tests have norms and is a formal assessment of the pupilÊs progress.

Alternative Assessment

Measurements can also be achieved through alternative assessments. Known also as the performance or authentic assessment, this assessment often occurs from time to time, and require pupils to generate products or to perform and not responding to the format of multiple-choice objectives or questions which are true or false. A few examples of alternative assessment are: essays,


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journals, assignments, group projects, portfolios, observations, interviews, examination, selfassessment and peer and class presentations. Detection evaluation

Evaluation is defined as the detection of an assessment conducted informally by parents, teachers, caregivers through observation or checklist for children under the age of 7 years. Done to collect information for record keeping process and also help in the detection during the earlier stage. In addition, the evaluation of detection is also used to help teachers and parents to be able to evaluate and track the progress of children.

Diagnostic Evaluation

Diagnostic evaluation is a process of evaluation of the child to diagnose the inability of a child. The diagnostic evaluation of children is usually done by a psychiatrist or teacher. It aims to make a preliminary assessment of the inability of the child and gather information for the placement of children according to their capabilities, and also to provide information to parents, teachers and paediatricians. It is important to determine the inability of children so that teachers are able to plan their teaching and learning methods.

Development Assessment

According to Rozumah Baharuddin (1984), the development is the change that happens in terms of quality but cannot be measured in quantity, although it can be seen by comparing the properties of the former with the nature of one which is being formed. This process is a complicated process that takes place directly from the early childhood to the end of life. Assessment of child development is measured in five aspects: evaluation aspects of motor, language, mental, social and self-management. The main purpose of evaluation is to assess the level of development of a child's development in terms of the development, whether there is a problem in achieving the level of development, or he/she cannot develop as a normal child. The assessment is also intended to detect deficiencies and advantages found in the children to improve development of children, to intervene early if the there is a deficiency found inand the child.

Performance Evaluation

According to the ÂKamus Dewan BahasaÊ, this evaluation is the process of assessing the achievements made on what has been achieved

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otherwise known as performance. Performance assessment is usually performed on children at age 7 years and above by the teacher to determine whether the pupil needs inclusive learning or not. This assessment is conducted to evaluate the efficiency and skills through observation and written tests. This assessment is divided into two types, somantic assessment and formative assessment. The purpose of this performance evaluation is carried out to assess the level of achievement of children, and analyse the strengths and weaknesses of the pupil. It enables teachers to know whether the objective of teaching and learning module had been achieved by the pupils or not, also to identify the learning needs of pupils and make the pupilsÊ placements according to their needs. Psychological Assessment

This assessment is intended to help evaluators control the behaviour of children with special needs in terms of mental, emotional and behavioural problems. The methods most commonly used for normal children may not be able to help the development of their behavioural, emotional, and mental needs. Thus the evaluators have to master themselves in the knowledge and skills to enable them to identify the special needs, and will that be able to provide treatment and early intervention fits well to help assess, investigate and make observations about the behaviour of children with special needs.

Assessments can and should be done anywhere preferably in small groups either at the playground, or done individually where teachers just need to talk to the pupils to get the information and feedback, or in a large group when children are doing leisure activities in the playground. We suggest that the assessment be made while the child is learning to apply mathematics to solve a problem in their daily activities. Evaluation should also be periodic and consistent with the child's learning. Example 1 describes how assessment can be conducted. This type of group interaction will normally occur in a kindergarten class.


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The above evaluation involved pupils in a group and the questions posed are the basis of mathematics. Through this assessment, the teacher can assess a pupilÊs agility in responding to the questions given on the spot. Pupils will not feel pressured by this group interview evaluation. The teacher's role in making an assessment of the pupils is a responsibility that should be noted accordingly. Teachers must be alert in observing and listening to the answers of the pupils which gives a high significant impact on the results of

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analysis. The use of audio and video recordings during the assessment is also encouraged and recommended. A planned and systematic observation with the aid of technology (audio and video) helps to make a valid and objective assessment. Moreover, it can be a proof to the parents whenever assessments are conducted on their children. In example 1 above, the teacher should observe the pupilÊs behaviour when counting the number of pupils in the classroom to ensure that the pupil calculate in a correct manner. The teacher should take note that the pupil doing the counting is using his/her finger and pointing to the correct person that he/she is counting.

ACTIVITY 10.2 As a teacher who teaches children in their early childhood education, how do you report the achievement of children to their parents?

10.3

ASSESSMENT STRATEGY

Assessment strategies during the learning of mathematics in early childhood education are meant to enhance the understanding of mathematics and promote skills that will benefit pupils throughout their lives. The ability to see the big picture, develop oral and written reports and the ability to work with their friends are skills that are encouraged. Table 10.3 displays the types of strategies that can be applied when making assessment. Every type of evaluation has its unique advantages and some combine more than one type of strategy. There are some strategies which are unsuitable for young children because they are not yet able to write well and interact smoothly yet. Assessment Strategy

Classroom Presentation

The presentation in the classroom is an evaluation strategy that requires pupils to express what they have learned, choose and submit examples of their work and arranged for topic to be presented. It can be a basic provision for project evaluation or pupil essays.

Conference

Conference is a formal or informal meetings between a teacher and a pupil for the purpose of exchanging information and sharing ideas. A conference may be held to explore ways a pupil


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thinks and recommend measures that will take place; assess a pupil's level of understanding of a particular concept or procedure; review and clarify the extent that pupils already know and have done. Essay

Exhibition/Demonstration

Essay is a written sample in which pupils build an answer to a question, topic, or a brief statement, and supply supporting documents or arguments. Essay allow teachers to evaluate, analyse and synthesise the information given by the pupils. Exhibition/demonstration is a presentation in a public environment, in which pupils describe and apply a process, procedures, etc., in a concrete way to demonstrate achievement of the individualÊs skills and specific knowledge.

Interview

Interviews are a face-to-face conversation in which the teacher and pupil interact with each other to investigate and share their knowledge and understanding of a topic or problem, and can be used by teachers to explore the pupils' thinking, assess the pupilÊs understanding of a concept or procedure and gather information, seek clarification, position, and probe for motivation.

Learning Log

Learning log is a continuous record kept by the pupils and recording of what he was doing or thinking when they perform certain tasks. It can be used to assess pupilsÊ progress and developments from time to time.

Observation

Observation is a process of systematically observing and recording the pupils while they work, for the purpose of programming and instructional decisions. Observations can occur at any time and in any setting. It provides information on a pupilÊs strengths and weaknesses, learning styles, interests, and attitudes.

Portfolio

Portfolios are collections of pupilsÊ work samples, which is focused, selective, reflective, and collaborative. It offers visual demonstration of a pupilÊs achievement, ability, strengths, weaknesses, knowledge, and specific skills from time to time and in a variety of contexts.

Verbal Questions and Answers

In the verbal question and answer strategies, teachers ask questions and pupils respond orally, not in writing. This strategy helps teachers to determine whether pupils understand what is being,

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or has been forwarded, and help pupils to extend their thinking, generate ideas, or solve problems. Quiz, Test and Examination

Journal Response

Quiz, test, or exam requires pupils to respond spontaneously to demonstrate their knowledge (oral or written) or skills (eg, through performance). Quizzes are usually short; examination usually takes a longer time. Quizzes, tests, or examinations can be customised for exceptional pupils and to repeat the lesson and redo the test. Journal response is the pupilsÊ personal records containing writing material, reflective responses to material read, sees, heard, or discussed. The journal can be used as an assessment tool in all subject areas.

Response Options

Some of the quizzes, tests, and exams, selected response requires pupils to correctly identify a response strategy. It can take the form of multiple choice or false or true format. Selected response is formal that commonly uses procedures to collect objective evidence about the pupilÊs learning, particularly in memorisation and comprehension.

Pupil Self-Assessment

Self-assessment is a process where pupils gather information that reflects their own learning. It is the pupil's own assessment of personal progress in terms of knowledge, skills, processes, or attitudes. Self-assessment gives the pupils greater awareness and understanding of themselves as learners.

Implementation of Duties

Implementation of task includes assignments, homework, quizzes, class presentations and creating products. Job implementation will help teachers access pupilsÊ achievement by providing marks in point form.

PupilÊs Writing / Self-Assessment

PupilsÊ writing include diagrams, pictures and charts. It provides a way to record the ways of thinking and assessing pupilÊs achievement concept. Sample essay is easy to store and will form a larger evaluation that is the portfolio. Writing techniques in mathematics is possible through journals, investigation reports, problems of mathematics, problem solving, and later writing self-assessment.

: Adapted from: The Ontario Curriculum Unit Planner; Assessment Strategies Companion, © QueenÊs Printer for Ontario, 2002


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ACTIVITY 10.3 There are five evaluation strategies that are considered uncommon, namely: observation, interview, job performance, and pupils' writing portfolio. Elaborate each of these strategies.

10.4

ASSESSMENT RESULTS ANALYSIS

All data collected from each pupil needs to be analysed to compare the information to meet the standards. This allows the teachers to monitor instruction, evaluate teaching strategies, track pupilsÊ progress and identify pupils with special needs who need guidance or extra service. However, the instructions should be given according to the needs and mental strength of each individual and teachers need to constantly monitor and observe a child's development while trying to modify any of the techniques in adaptation to the individual pupil. The teacherÊs sensitivity in analysing assessment results will help improve the performance of each pupil to the optimum level. Among the advantages of conducting analysis on each of the assessment results are: (a)

PupilsÊ excellence in achievement can be reinforced and enhanced through challenging activities. Lack of pupilÊs achievement can be helped through proper guidance;

(b)

The development and progress of pupils can be monitored fairly through individual, group, and class learning process;

(c)

A pupil who has mastered the level of achievement should be supported and given more consolidation activities to improve achievement;

(d)

Pupils who have not mastered the level of achievement should be given guidance and support activities that are appropriate to the level of pupilÊs mastery; and

(e)

Teachers should also inform parents of their child's proficiency level to gain extra support at home.

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•

The assessment is designed to identify a wide range of intelligence and potential to be strengthened and improved.

•

Assessment plays an important role in the development of children's learning. Assessment must be tailored by age and level of children's minds

•

•

and conducted in stages. Among the 15 strategies of evaluation, five are more frequently used; the strategy evaluation observations, interviews, implementation of tasks, and the pupils' writing self-assessment and portfolios. Each assessment must be analysed in order to achieve high standards and to be fair to both sides, that is, the assessment that is made has to be rational and can help develop children's learning.

Alternative assessment

Response journal

Conference Learning log

Response option Self- assessment

Portfolio

Standard assessment

Psychological assessment

The Ontario Curriculum Unit Planner (2002). Assessment strategies companion. QueenÊs Printer for Ontario.

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E-mail your comment or feedback to [email protected]

OR 2.

Fill in the Print Module online evaluation form available on myVLE.

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HBEC2503 Mathematics in Early Childhood Edu

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